Arithmetic Means : Problems and Solutions

Definition: Arithmetic Mean

In a finite arithmetic progression, the terms between the first term and the last term are called the arithmetic means.

အဆုံးရှိ A.P ကိန်းစဉ် တစ်ခု၏ ရှေ့ဆုံးကိန်းနှင့် နောက်ဆုံးကိန်းကြားရှိ ကိန်းများအားလုံးကို arithmetic means ဟုခေါ်သည်။

If $\boldsymbol{u}_{1}, \boldsymbol{u}_{2}, \boldsymbol{u}_{3}, \ldots \boldsymbol{u}_{n-1}, \boldsymbol{u}_{n}$ is an A.P., then $\boldsymbol{u}_{2}, \boldsymbol{u}_{3}, \ldots, \boldsymbol{u}_{n-1}$ are called arithmetic means. The arithmetic mean between two numbers $x$ and $y$ is given by

$\begin{array}{|l|}\hline \text {A.M} = \dfrac{x+y}{2}\\ \hline \end{array}$

Exercises

1. Find the A.M. between
   (a) $-3$ and 3.
   (b) $2-\sqrt{2}$ and $2+\sqrt{2}$.
   (c) $\log 3$ and $\log 12$.





$\begin{aligned} &\text{The A.M. between} x \text{ and } y= \frac{x+y}{2} \\\\ \text{(a) } &\text{The A.M. between } -3 \text{ and } 3 \\\\ =&\frac{-3+3}{2} \\\\ =& 0 \\\\ \text{(b) } &\text{The A.M. between } 2-\sqrt{2} \text{ and } 2+\sqrt{2}\\\\ =&\frac{2-\sqrt{2}+2+\sqrt{2}}{2} \\\\ =& 2 \\\\ \text{(c) } &\text{The A.M. between } \log 3 \text{ and } \log 12\\\\ =&\frac{\log 3+\log 12}{2} \\\\ =& 2 \\\\ =&\frac{1}{2} \log 36 \\\\ =&\log \sqrt{36} \\\\ =&\log 6 \end{aligned}$



2. Insert three arithmetic means between $-5$ and $19$ .





Let the required arithmetic means be $x_{1}, x_{2}$ and $x_{3}$. $\begin{aligned} \therefore\ -5, x_{1},\ & x_{2},\ x_{3},\ 19 \text { is an A.P. } \\\\ \therefore\ a &=-5 \\\\ u_{5}&=19 \\\\ a+4 d&=19 \\\\ -5+4 d&=19 \\\\ 4 d&=24\\\\ d&=6 \\\\ x_{1}&=a+d=1 \\\\ x_{2}&=a+2 d=7 \\\\ x_{3}&=a+3 d=13 \end{aligned}$



3. Insert five arithmetic means between $p+q$ and $19 p-11 q$.





Let the required arithmetic means be $x_{1}, x_{2}, x_{3}, x_{4}$ and $x_{5}$. $\therefore p+q, x_{1}, x_{2}, x_{3}, x_{4}, x_{5}, 19 p-11 q$ is an A.P. Let the first term be $a$ and the common difference be $d$. $\begin{aligned} \therefore\ a &=p+q \\\\ u_{7} &=19 p-11 q \\\\ a+6 d &=19 p-11 q\\\\ \therefore\ 6 d &=18 p-12 q \\\\ d &=3 p-2 q \\\\ \therefore\ x_{1} &=a+d=4 p-q \\\\ x_{2} &=a+2 d=7 p-3 q \\\\ x_{3} &=a+3 d=10 p-5 q \\\\ x_{4} &=a+4 d=13 p-7 q \\\\ x_{5} &=a+5 d=16 p-9 q \end{aligned}$



4. If five arithmetic means are inserted between $-10$ and $116$, what is the third A.M.?





Let the fine A.Ms between $-10$ and $n_{6}$ be $x_{1}, x_{2}, x_{3}, x_{4}, x_{5}$. $\therefore-10, x_{1}, x_{2}, x_{3}, x_{4}, x_{5}, 116$ is an A.P. Let the first term be $a$ and the common difference be $d$. $\begin{aligned} \therefore\ a &=-10 \\\\ u_{7} &=116 \\\\ a+6 d &=116 \\\\ \therefore\ 6 d &=126 \\\\ d &=21 \\\\ \therefore\ x_{3} &=a+3 d \\\\ &=53 \end{aligned}$



5. If $n$ arithmetic means are inserted between $a$ and $b$, show that the common difference of the A.P. is $\dfrac{b-a}{n+1}$.





Let the $n$ arithmetic means between $a$ and $b$ be $x_{1}, x_{2}, x_{3}, \ldots, x_{n} .$ $\therefore\ a, x_{1}, x_{2}, x_{3}, \ldots, x_{n}, b$ is an A.P. Let the common difference be $d$. $\begin{aligned} u_{n+2} &=b \\\\ a+(n+2-1) d &=b \\\\ (n+1) d &=b-a \\\\ \therefore\ d &=\dfrac{b-a}{n+1} \end{aligned}$



6. If $n$ arithmetic means are inserted between $20$ and $80$ such that the ratio of first mean to the last mean is $1: 3$, find the value of $n$.





Let the $n$ arithmetic means between 20 and 80 be $x_{1}, x_{2}, x_{3}, \ldots, x_{n}$. $\therefore 20, x_{1}, x_{2}, x_{3}, \ldots, x_{n}, 80$ is an A.P. Let the first termbe $a$ and the common differenee be $d$. $\begin{aligned} \therefore \ a &=20 \\\\ u_{n+2} &=80\\\\ a+(n+2-1) d &=80 \\\\ 20+(n+1) d &=80 \\\\ d &=\dfrac{60}{n+1} \\\\ \dfrac{x_{1}}{x_{n}} &=\dfrac{1}{3} \\\\ \dfrac{a+d}{a+n d} &=\dfrac{1}{3}\\\\ 3 a+3 d &=a+n d \\\\ 2 a &=(n-3) d \\\\ 40 &=(n-3) \dfrac{60}{n+1} \\\\ 2 n+2 &=3 n-9 \\\\ n &=11 \end{aligned}$



7. If the A.M. between $p^{\text {th }}$ and $q^{\text {th }}$ terms of an A.P. be equal to the A.M. between $r^{\text {th }}$ and $s^{\text {th }}$ terms of the A.P., show that $p+q=r+s$.





$\begin{aligned} \text { A.M. between } u_{p} \text { and } u_{q} &=\dfrac{u_{p}+u_{q}}{2} \\\\ &=\dfrac{a+(p-1) d+a+(q-1) d}{2} \\\\ &=\dfrac{2 a+(p+q-2) d}{2} \\\\ &=a+\dfrac{1}{2}(p+q-2) d \\\\ \text { A.M. between } u_{r} \text { and } u_{s} &=\dfrac{u_{r}+u_{s}}{2} \\\\ &=\dfrac{a+(r-1) d+a+(s-1) d}{2} \\\\ &=\dfrac{2 a+(r+s-2) d}{2} \\\\ &=a+\dfrac{1}{2}(r+s-2) d\\\\ \text {By the problem}, \quad \quad \quad&\\\\ a+\dfrac{1}{2}(p+q-2) d &=a+\dfrac{1}{2}(r+s-2) d \\\\ \therefore p+q &=r+8 \end{aligned}$



8. If $x, y, z$ are in A.P. and $A_{1}$ is the A.M. between $x$ and $y$, and $A_{2}$ is the A.M. between $y$ and $z$, prove that the A.M. between $A_{1}$ and $A_{2}$ is $y$.





$\begin{aligned} x, y, z \text{ are in A.P}&\\\\ \therefore\quad y&=\dfrac{x+z}{2}\\\\ \text{ A.M between } x \text{ and } y&=\dfrac{x+y}{2}\\\\ \therefore\quad A_{1}&=\dfrac{x+y}{2}\\\\ \text{ A.M between } y \text{ and } z&=\dfrac{y+z}{2}\\\\ \therefore\quad A_{2}&=\dfrac{y+z}{2}\\\\ \text{ A.M between } A_{1} \text{ and } A_{2}&=\dfrac{A_{1}+A_{2}}{2}\\\\ &=\dfrac{1}{2}\left(\dfrac{x+y}{2}+\dfrac{y+z}{2}\right) \\\\ &=\dfrac{1}{2}\left(\dfrac{x+z}{2}+y\right) \\\\ &=\dfrac{1}{2}(y+y) \\\\ &=\dfrac{1}{2}(2 y) \\\\ &=y \end{aligned}$



9. If $x$ is the A.M. between $a$ and $b$, show that $\dfrac{x+2 a}{x-b}+\dfrac{x+2 b}{x-a}=4$.





$\begin{aligned} x \text{ is the }& \text{ A.M. between } a \text{ and } b.\\\\ \therefore \quad x&=\dfrac{a+b}{2}\quad\quad \\\\ \dfrac{x+2 a}{x-b}+\dfrac{x+2 b}{x-a} &=\dfrac{\dfrac{a+b}{2}+2 a}{\dfrac{a+b}{2}-b}+\dfrac{\dfrac{a+b}{2}+2 b}{\dfrac{a+b}{2}-a}\\\\ &=\dfrac{5 a+b}{a-b}+\dfrac{a+5 b}{b-a} \\\\ &=\dfrac{5 a+b}{a-b}+\dfrac{-a-5 b}{a-b} \\\\ &=\dfrac{4 a-4 b}{a-b} \\\\ &=\dfrac{4(a-b)}{a-b}\\\\ &=4 \end{aligned}$

Geometric Progression : Problems and Solutions -Part (1)

Definition: Geometric Progression

A geometric progression is a sequence in which the ratio of each term to the one before it, is a constant.
ကိန်းစဉ်တစ်ခု၏ နီးစပ် (ကပ်လျက်) ကိန်းနှစ်လုံး (နောက်ကိန်းနှင့် ရှေ့ကိန်း)၏ အချိုးသည် ကိန်းသေဖြစ်လျှင် ၎င်းကိန်းစဉ်ကို geometric progression ဟုခေါ်သည်။
This ratio is called the common ratio and is denoted by $r$.
If If $u_{1}, u_{2}, u_{3}, \ldots, u_{n-1}, u_{n}$ is an G.P., then $\dfrac{u_{2}}{u_{1}}=\dfrac{u_{3}}{u_{2}}=\ldots=\dfrac{u_{n}}{u_{n-1}}=$ constant.
$\dfrac{u_{n}}{u_{n-1}}=r$ and $u_{n}=u_{n-1} \cdot r$ where $r$ is called the common ratio.
If the first term is $a$, the the sequence may be $a, ar, ar^2, ar^3, \ldots$.
Hence the $n^{\text{th}}$ term is,

$\begin{array}{|l|}\hline u_n = ar^{n-1}\\ \hline \end{array}$

Definition: Geometric Mean (G.M.)

In a finite geometric progression, the terms between the first term and the last term are called the geometric means.
အဆုံးရှိ geometric progression တစ်ခု၏ ရှေ့ဆုံးကိန်းနှင့် နောက်ဆုံးကိန်းကြားရှိ ကိန်းများအားလုံးကို geonetric means (G.M) ဟုခေါ်သည်။
If $u_{1}, u_{2}, u_{3}, \ldots u_{n-1}, u_{n}$ is an G.P., then
$u_{2}, u_{3}, \ldots, u_{n-1}$ are called geometric means.
The geometric mean between two numbers $x$ and $y$ is given by

$\begin{array}{|l|}\hline \text{G.M } = \sqrt{xy}\\ \hline \end{array}$

Exercises

1. Find
   (a) the common ratio,
   (b) the $10^{\text {th }}$ term and
   (c) the $n^{\text {th }}$ term of the following G.P.
   (i) $4,2,1, \dfrac{1}{2}, \ldots$
   (ii) $-2,4,-8,16, \ldots$
   (iii) $5,20,80,320, \ldots$
   (iv) $\dfrac{1}{3},-\dfrac{1}{9}, \dfrac{1}{27},-\dfrac{1}{81}, \ldots$
   (v) $\dfrac{8}{9}, \dfrac{4}{3}, 2,3, \ldots$
   (vi) $x^{5}, x^{4} y, x^{3} y^{2}, x^{2} y^{3}, \ldots$





=======Sol 1=======



2. If $3, x, y, z, w$ and 3072 are consecutive terms of a G.P., find the value of $x$, $y, z$ and $w$.





$3, x, y, z, w, 3072$ is a G.P. $\begin{aligned} & \\ \therefore\ &a=3 \text{ and }\\\\ &u_{6}=3072\\\\ &a r^{5}=3072\\\\ &3 r^{5}=3072\\\\ &r^{5}=1024\\\\ & r=(4)^{5} \\\\ & r= 4\\\\ &\therefore\ x=u_{2}=a r=-12 \\\\ & y=u_{3}=a r^{2}=48 \\\\ & z=u_{4}=a r^{3}=-192 \\\\ & w=u_{5}=a r^{4}=768\\\\ \end{aligned}$



3. The second term of a G.P. is 64 and the fifth term is 27 . Find the first 6 terms of the G.P.





In a G.P., $u_{2}=64$ $a r=64 \ldots(1)$ $u_{5}=27$ $a r^{4}=27 \ldots(2)$ $(2) \div(1)$ $\dfrac{a r^{4}}{a r}=\dfrac{27}{64}$ $r^{3}=\left(\dfrac{3}{4}\right)^{3}$ $r=\dfrac{3}{4}$ $\therefore\ a\left(\dfrac{3}{4}\right)=64$ $\therefore\ a=\dfrac{256}{3}$ $\begin{aligned} \therefore\ u_{1} &=a=\dfrac{256}{3} \\\\ u_{2} &=a r=64 \\\\ u_{3} &=a r^{2}=48 \\\\ u_{4} &=a r^{3}=36 \\\\ u_{5} &=a r^{4}=27 \\\\ u_{6} &=a r^{5}=\dfrac{81}{4} \end{aligned}$



4. Find the $10^{\text {th }}$ term of the G.P. $a^{5}, a^{4} b, a^{3} b^{2}, a^{2} b^{3}, \ldots$. Which term of the G.P. is $\dfrac{b^{20}}{a^{15}} ?$





$a^{5}$, $a^{4} b$, $a^{3} b^{2}$, $a^{2} b^{3}$, $\ldots$ is a GP. Let the first term be $A$ and the common ratio be $r$. Then, $\begin{aligned} &A=a^{5} \\\\ &r=\dfrac{a^{4} b}{a^{5}}=\dfrac{b}{a} \end{aligned}$ Let $\dfrac{b^{20}}{a^{15}}$ be the $n^{\text {th }}$ term. $\begin{aligned} \therefore u_{n} &=\dfrac{b^{20}}{a^{15}} \\\\ A r^{n-1} &=\dfrac{b^{20}}{a^{15}} \\\\ a^{5}\left(\dfrac{b}{a}\right)^{n-1} &=\dfrac{b^{20}}{a^{15}} \\\\ \left(\dfrac{b}{a}\right)^{n-1} &=\left(\dfrac{b}{a}\right)^{20}\\\\ \therefore\ n-1 &=20 \\\\ n &=21 \\\\ \therefore\ u_{21}&=\frac{b^{20}}{a^{15}} \end{aligned}$



5. The $4^{\text {th }}$ term of the G.P is 3 and the sixth term is 147 . Find the first three terms of the two possible geometric progressions.





In a G.P., $\begin{aligned} u_{4}&=3\\\\ a r^{3}&=3 \ldots(1)\\\\ u_{6}&=147\\\\ a r^{5}&=147 \ldots(2)\\\\ (2) \div &(1)\\\\ \dfrac{a r^{5}}{a r^{3}}&=\dfrac{147}{3}\\\\ r^{2} &=49 \\\\ r &=\pm 7 \\\\ \text { When } r &=-7 \\\\ a(-7)^{3} &=3 \\\\ a &=-\dfrac{3}{343} \\\\ \therefore\ u_{1}&=a=-\dfrac{3}{343} \\\\ u_{2}&=a r=\dfrac{3}{49} \\\\ u_{3}&=a r^{2}=-\dfrac{3}{7}\\\\ \text { When } r &=7 \\\\ a(7)^{3} &=3 \\\\ a &=\dfrac{3}{343} \\\\ \therefore \quad u_{1} &=a=\dfrac{3}{343} \\\\ u_{2} &=a r=\dfrac{3}{49} \\\\ u_{2} &=a r^{2}=\dfrac{3}{7} \end{aligned}$



6. The product of the first three terms of the G.P. is 1 and the product of the third, fourth and the fifth term is $11 \dfrac{25}{64}$. Find the fifth term of the G.P.





Let the given G.P. be $a, a r, a r^{2}, a r^{3}, a r^{4}, \ldots$ By the problem, $\begin{aligned} a \cdot a r \cdot a r^{2} &=1 \\\\ (a r)^{3} &=1 \\\\ a r &=1 \\\\ a &=\dfrac{1}{r}\\\\ a r^{2} \cdot a r^{3} \cdot a r^{4} &=11 \dfrac{25}{64} \\\\ a^{3} r^{9} &=\dfrac{729}{64} \\\\ \dfrac{1}{r^{3}} \cdot r^{9} &=\dfrac{729}{64} \\\\ r^{6} &=\dfrac{729}{64} \\\\ r^{6} &=\left(\pm \dfrac{3}{2}\right)^{6}\\\\ \therefore r &=\pm \dfrac{3}{2} \\\\ a &=\dfrac{1}{r} \\\\ \therefore a &=\pm \dfrac{2}{3} \\\\ u_{5} &=a r^{4} \\\\ &=\pm \dfrac{2}{3}\left(\pm \dfrac{3}{2}\right)^{4} \\\\ &=\pm \dfrac{27}{8} \end{aligned}$



7. Find two different values of $x$, so that $-\dfrac{3}{2}, x,-\dfrac{8}{27}$ will be a G.P.





$\begin{aligned} \text { If }-\dfrac{3}{2}, x,-\dfrac{8}{27} &\text { is a G.P., then } \\\\ \dfrac{x}{\left(-\dfrac{3}{2}\right)}&=\dfrac{\left(-\dfrac{8}{27}\right)}{x} \\\\ \therefore\ x^{2}&=\dfrac{4}{9} \\ x&=\pm \dfrac{2}{3} \end{aligned}$



8. Find which term of the G.P. $\dfrac{8}{9}$, $\dfrac{4}{3} \sqrt{\dfrac{2}{3}}$, $\dfrac{4}{3}, \ldots$ is $\sqrt{6}$.





$\dfrac{8}{9}$, $\dfrac{4}{3} \sqrt{\dfrac{2}{3}}$, $\dfrac{4}{3}$, $\ldots$, $\sqrt{6}$ is a G.P. $\begin{aligned} \therefore\ a &=\dfrac{8}{9} \\\\ r &=\dfrac{\dfrac{4}{3}}{\dfrac{4}{3} \sqrt{\dfrac{2}{3}}}=\sqrt{\dfrac{3}{2}}\\\\ \text{Let } u_{n}&=6, \text{then }\\\\ \therefore\ a^{n-1}&=\sqrt{6}\\\\ \dfrac{8}{9}\left(\sqrt{\dfrac{3}{2}}\right)^{n-1} &=\sqrt{6} \\\\ \left(\sqrt{\dfrac{3}{2}}\right)^{n-1} &=\dfrac{9 \sqrt{6}}{8} \\\\ &=\sqrt{\dfrac{9 \times 9 \times 6}{8 \times 8}} \\\\ &=\sqrt{\dfrac{3^{5}}{2^{5}}} \\\\ &=\left(\sqrt{\dfrac{3}{2}}\right)^{5}\\\\ \therefore\ n-1 &=5 \\\\ n &=6 \\\\ \therefore\ u_{6} &=\sqrt{6} \end{aligned}$



9. If $a, b, c, d$ is a G.P., show that $a^{2}-b^{2}$, $b^{2}-c^{2}$, $c^{2}-d^{2}$ is also a G.P.





$a, b, c, d$ is a G.P. Let the common ratio be $r$, then $\begin{aligned} a&=a \\\\ b&=a r \\\\ c&=a r^{2} \\\\ d&=a r^{3}\\\\ \dfrac{b^{2}-c^{2}}{a^{2}-b^{2}} &=\dfrac{a^{2} r^{2}-a^{2} r^{4}}{a^{2}-a^{2} r^{2}} \\\\ &=\dfrac{r^{2}\left(a^{2}-a^{2} r^{2}\right)}{a^{2}-a^{2} r^{2}} \\\\ &=r^{2} \\\\ \dfrac{c^{2}-d^{2}}{b^{2}-c^{2}} &=\dfrac{a^{2} r^{4}-a^{2} r^{6}}{a^{2} r^{2}-a^{2} r^{4}} \\\\ &=\dfrac{r^{2}\left(a^{2} r^{2}-a^{2} r^{4}\right)}{a^{2} r^{2}-a^{2} r^{4}} \\\\ &=r^{2}\\\\ \therefore\ \dfrac{b^{2}-c^{2}}{a^{2}-b^{2}}&=\dfrac{c^{2}-d^{2}}{b^{2}-c^{2}} \end{aligned}$ $\therefore\ a^{2}-b^{2}, b^{2}-c^{2}, c^{2}-d^{2}$ is a G.P.



10. If $a, b, c, d$ is a G.P., show that
    (i) $\dfrac{b+c}{c+d}=\dfrac{a+c}{b+d}$.
    (ii) $(a+d)(b+c)-(a+c)(b+d)=(b-c)^{2}$.





$a, b, c, d$ is $a G . P .$ Let the common ratio be $r$, then $\begin{aligned} a&=a \\\\ b&=a r \\\\ c&=a r^{2} \\\\ d&=a r^{3}\\\\ \text { (i) }\quad \dfrac{b+c}{c+d} &=\dfrac{a r+a r^{2}}{a r^{2}+a r^{3}} \\\\ &=\dfrac{a r+a r^{2}}{r\left(a r+a r^{2}\right)} \\\\ &=\dfrac{1}{r} \\ \dfrac{a+c}{b+d} &=\dfrac{a+a r^{2}}{a r+a r^{3}} \\\\ &=\dfrac{a+a r^{2}}{r\left(a+a r^{2}\right)} \\\\ &=\dfrac{1}{r}\\\\ \therefore\ \dfrac{b+c}{c+d}&=\dfrac{a+c}{b+d} \\\\ \text { (ii) }\quad\quad\quad &(a+d)(b+c)-(a+c)(b+d) \\\\ =\ & a b+a c+b d+c d-a b-a d-b c-c d \\\\ =\ & a c+b d-a d-b c \\\\ =\ & a\left(a r^{2}\right)+(a r)\left(a r^{3}\right)-a\left(a r^{3}\right)-(a r)\left(a r^{2}\right) \\\\ =\ &(a r)^{2}+\left(a r^{2}\right)^{2}-2 a^{2} r^{3} \\\\ =\ &(a r)^{2}-2(a r)\left(a r^{2}\right)+\left(a r^{2}\right)^{2} \\\\ =\ & b^{2}-2 b c+c^{2} \\\\ =\ &(b-c)^{2} \end{aligned}$



11. If $a, b, c$ is an A.P. and $x, y, z$ is a G.P., show that $x^{b-c} y^{c-a} z^{a-b}=1$.





$a, b, c$ is an $A P$. Let $d$ be the common difference. $\begin{aligned} \therefore a =\ &a \\\\ b &= a+d \\\\ c &= &a+2 d \end{aligned}$ $x, y, z$ is a G.P Let $r$ be the common ratio. $\begin{aligned} \therefore x =\ &x \\\\ y =\ &x r \\\\ z =\ &x r^{2} \\\\ & x^{b-c} y^{c-a} z^{a-b} \\\\\ =\ &x^{a+d-a-2 d}(x r)^{a+2 d-a}\left(x r^{2}\right)^{a-a-d} \\\\ =\ &x^{-d} \cdot x^{2 d} \cdot r^{2 d} \cdot x^{-d} \cdot r^{-2 d} \\\\ =\ &x^{0} r^{0} \\\\ =\ &1 \end{aligned}$



12. In a G.P. the product of any three consecutive terms is $512$ . When 8 is added to the first term and $6$ to the second, then the terms form an A.P. Find the terms of the G.P.





Let $a$, ar, ar be ginen G.P., then $\begin{aligned} &a \cdot a r \cdot a r^{2}=512 \\\\ &(a r)^{3}=8^{3} \\\\ &a r=8 \\\\ &a=\dfrac{8}{r} \end{aligned}$ $\therefore$ The terms of given G.P. are $\dfrac{8}{r}, 8,8 r .$ By the problem, $\begin{aligned} \dfrac{8}{r}+8,(8+6), 8 r & \text { is an AP. } \\\\ \therefore 14-\dfrac{8}{r}-8 &=8 r-14 \\\\ 6-\dfrac{8}{r}&=8 r-14 \\\\ 6 r-8&=8 r^{2}-14 r \\\\ 8 r^{2}-20 r+8&=0 \\\\ 2 r^{2}-5 r+2&=0 \\\\ (2 r-1)(r-2)&=0\\\\ \end{aligned}$ $\therefore r=\dfrac{1}{2}$ or $r=2$. When $r=\dfrac{1}{2}$ the terms of G.P ane $16,8,4$. When $r=2$, the terms of G.P. are $4,8,16$.



13. In a G.P. the product of any three consecutive terms is $216$ . When 1 is added to the first term and $2$ to the second, then the terms form an A.P. Find the terms of the G.P.





Let $a$, ar, ar be ginen G.P., then $\begin{aligned} &a \cdot a r \cdot a r^{2}=216 \\\\ &(a r)^{3}=6^{3} \\\\ &a r=8 \\\\ &a=\dfrac{6}{r} \end{aligned}$ $\therefore$ The terms of given G.P. are $\dfrac{6}{r}, 6,6 r .$ By the problem, $\begin{aligned} \dfrac{6}{r}+1,(6+2), 6 r & \text { is an AP. } \\\\ \therefore 8-\dfrac{6}{r}-1 &=6 r-8 \\\\ 7-\dfrac{6}{r}&=6 r-8 \\\\ 7 r-6&=6 r^{2}-8 r \\\\ 6 r^{2}-15 r+6&=0 \\\\ 2 r^{2}-5 r+2&=0 \\\\ (2 r-1)(r-2)&=0\\\\ \end{aligned}$ $\therefore r=\dfrac{1}{2}$ or $r=2$. When $r=\dfrac{1}{2}$ the terme of G.P ane $12,6,3$. When $r=2$, the terms of G.P. are $3,6,12$.



14. $8, x, y$ are three consecutive terms of an A.P. while $x, y, 36$ are three consecutive terms of a G.P., find the possible values of $x$ and $y$.





$8, x, y$ is an A.P. $\begin{aligned} \therefore\ x -8 & =y-x \\\\ y &=2 x-8 \\\\ &=2(x-4)--(1) \\\\ x, y &, 36 \text { is a G.P. } \\\\ \dfrac{y}{x} &=\dfrac{36}{y} \\\\ \therefore\ y^{2} &=36 x-(2) \end{aligned}$ Substituting $y=2 x-8$ in equation (2), $\begin{aligned} 4(x-4)^{2} &=36 x \\\\ x^{2}-8 x+16 &=9 x \\\\ x^{2}-17 x+16 &=0 \\\\ \therefore\ (x-1)(x-16) &=0 \\\\ x=1 \text { or } x &=16 \end{aligned}$ When $x=1, y=2(1-4)=-6$ When $x=16, y=2(16-4)=24$



15. In a G.P., whose first term is positive, the sum of the first and the third term is $\dfrac{13}{9}$ and the product of the second and fourth term is $\dfrac{16}{81}$. Find the common ratio and the $6^{\text {th }}$ term.





Let $a$, ar, $a r^{2}$ be given GI.P., where $a>0$. By the problem, $\begin{aligned} a+a r^{2}&=\dfrac{13}{9} \ldots (1)\\\\ ar \cdot ar^{2}&=\dfrac{16}{81}\\\\ \left(a r^{2}\right)^{2}&=\dfrac{16}{81}\\\\ \therefore\ a r^{2}&=\dfrac{4}{9} \left(\because a>0 \text { and } r^{2}>0\right)\ldots(2) \end{aligned}$ Substituting $a r^{2}=\dfrac{4}{9}$ in equation $(1)$, $\begin{aligned} a+\dfrac{4}{9} &=\dfrac{13}{9} \\\\ a &=1\\\\ \therefore 1\left(r^{2}\right) &=\dfrac{4}{9} \\\\ r &=\pm \dfrac{2}{3} \end{aligned}$ When $a=1$, and $r=-\dfrac{2}{3}$, $\begin{aligned} u_{6} &=a r^{5} \\\\ &=1\left(-\dfrac{2}{3}\right)^{5} \\\\ &=-\dfrac{32}{243} \end{aligned}$ When $a=1$, and $r=\dfrac{2}{3}$ $\begin{aligned} u_{6} &=a r^{5} \\\\ &=1\left(\dfrac{2}{3}\right)^{5} \\\\ &=\dfrac{32}{243} \end{aligned}$



16. If $\log _{3} 2$, $\log _{4} x$, $\log _{2} 81$ is a G.P. then find the possible values of $x$.





$\log _{3} 2, \log _{4} x, \log _{2} 81$ is a G.P. $\begin{aligned} \therefore\ \dfrac{\log _{4} x}{\log _{3} 2} &=\dfrac{\log _{2} x}{\log _{4} x} \\\\\ \therefore\ \left(\log _{4} x\right)^{2} &=\log _{2} 81 \times \log _{3} 2 \\\\\ &=4 \log _{2} 3 \times \dfrac{1}{\log _{2} 3}\\\\\ \left(\log _{4} x\right)^{2}&=4 \\\\\ \therefore\ \log _{4} x&=\pm 2 \\\\\ \therefore\ x &=4^{-2} \text { or } x=4^{2} \\\\\ x &=\dfrac{1}{16} \text { or } x=16 \end{aligned}$



17. If $\dfrac{1}{b+a}, \dfrac{1}{2 b}$ and $\dfrac{1}{b+c}$ are in A.P., prove that $a, b$ and $c$ are in G.P.





$\begin{aligned} \dfrac{1}{b+a}, \dfrac{1}{2 b}, \dfrac{1}{b+c}\ & \text {is an A.P } \\\\\ \therefore \dfrac{1}{2 b}-\dfrac{1}{b+a} &=\dfrac{1}{b+c}-\dfrac{1}{2 b} \\\\\ \dfrac{b+a-2 b}{2 b(b+a)} &=\dfrac{2 b-b-c}{2 b(b+c)} \\\\\ \dfrac{a-b}{a+b} &=\dfrac{b-c}{b+c}\\\\\ \text {By Componendo} &- \text {Dividendo}\\\\ \dfrac{(a-b)+(a+b)}{(a-b)-(a+b)} &=\dfrac{(b-c)+(b+c)}{(b-c)-(b+c)} \\\\\ -\dfrac{2 a}{2 b} &=-\dfrac{2 b}{2 c} \\\\\ \dfrac{a}{b} &=\dfrac{b}{c} \\\\\ \therefore \dfrac{b}{a} &=\dfrac{c}{b}\\\\\ \therefore a, b, c & \text { are in G.P.} \end{aligned}$



18. In a G.P., the first term exceeds the third term by $72$ and the sum of the second and third term is $36$ . Find the first term.





Let $a$, $ar$, $ar^{2}$ be given G.P. By the problem, $\begin{aligned} a-a r^{2}&=72 \\\\ a\left(1-r^{2}\right)&=72\ldots(1) \\\\ a r+a r^{2}&=36 \\\\ a r(1+r)&=36 \ldots(2)\\\\ (1) \div(2) & \\\\ \dfrac{a\left(1-r^{2}\right)}{\operatorname{ar}(1+r)} &=\dfrac{72}{3 b} \\\\ \dfrac{(1-r)(1+r)}{r(1+r)} &=2 \\\\ \therefore \quad 1-r &=2 r \\\\ 3 r &=1 \\\\ r &=\dfrac{1}{3}\\\\ \text{Substituting } r=\dfrac{1}{3}& \text{ in equation} (1),\\\\ a\left(1-\dfrac{1}{9}\right)=72 \\\\ \dfrac{8 a}{9}=72 \\\\ a=81 \end{aligned}$



19. The fourth term of a G.P. is $9$ and the ninth term is $2187$. Find the first 4 terms of the G.P.





$\begin{aligned} \text{In a G.P, } & \\\\ u_{4}&=9 \\\\ a r^{3}&=9\ldots(1) \\\\ u_{9}&=2187 \\\\ a{r}^{8}&=2187\ldots(2) \\\\ (2) \div(1),& \\\\ \dfrac{a r^{8}}{a r^{3}}&=\dfrac{2187}{9}\\\\ r^{5}&=243 \\\\ r^{5}&=3^{5} \\\\ r&=3 \end{aligned}$ Substituting $r=3$ in equation (1), $\begin{aligned} a\left(3^{3}\right)&=9 \\\\ a&=\dfrac{1}{3}\\\\ \therefore\ u_{1} &=a=\dfrac{1}{3} \\\\ u_{2} &=a r=1 \\\\ u_{3} &=a r^{2}=3 \\\\ u_{4} &=a^{3}=9 \end{aligned}$



20. The fourth term of a G.P. exceeds the third by $\dfrac{3}{44}$ and the third term exceeds the second term by $\dfrac{1}{22}$. Find the first term and the sixth term of the G.P.





In a G.P, $\begin{aligned} u_{4}-u_{3}&=\dfrac{3}{44} \\\\ a r^{3}-a r^{2}&=\dfrac{3}{44} \\\\ a r^{2}(r-1)&=\dfrac{3}{44}\ldots(1)\\\\ u_{3}-u_{2} &=\dfrac{1}{22} \\\\ {ar}^{2}-a r &=\dfrac{1}{22} \\\\ {ar}(r-1) &=\dfrac{1}{22}\ldots(2) \\\\ (1) \div(2), & \\\\ {ar}^{2}(r-1) & \\\\ {ar}(r-1) &=\dfrac{1}{\dfrac{3}{22}}\\\\ r &=\dfrac{3}{44} \times 22 \\\\ &=\dfrac{3}{2}\\\\ \end{aligned}$ Substituting $r=\dfrac{3}{2}$ in equation (2), $\begin{aligned} \dfrac{3 a}{2}\left(\dfrac{3}{2}-1\right) &=\dfrac{1}{22} \\\\ \dfrac{3 a}{4} &=\dfrac{1}{22} \\\\ a &=\dfrac{2}{33} \\\\ \therefore \quad u_{6} &=a r^{5} \\\\ &=\left(\dfrac{2}{33}\right)\left(\dfrac{3}{2}\right)^{5} \\\\ &=\dfrac{81}{176} \end{aligned}$



21. Three consecutive terms of a G.P. are $3^{2 x-1}, 9^{x}$ and $243$ . Find the value of $x$. If $243$ is the fifth term of the G.P., find the seventh term.





$3^{2 x-1}, 9^{x}, 243$ are in a G.P. $\begin{aligned} \therefore\ \dfrac{9^{x}}{3^{2 x-1}} &=\dfrac{243}{9^{x}} \\\\ \dfrac{3^{2 x}}{3^{2 x-1}} &=\dfrac{3^{5}}{3^{2 x}} \\\\ 3 &=3^{5-2 x} \\\\ \therefore \quad 5-2 x &=1 \\\\ x &=2\\\\ \therefore\ u_{3} &=3^{4-1}=27 \\\\ u_{4} &=9^{2}=81 \\\\ u_{5} &=243 \\\\ \therefore\ r &=\dfrac{81}{27} \\\\ &=3 \\\\ \therefore u_{7} &=a r^{6} \\\\ &=\left(a r^{4}\right) r^{2} \\\\ &=u_{5} \times r^{2}=243 \times 9=2187 \end{aligned}$



22. Find three numbers in a G.P. such that their sum is $42$ and their product is $512$.





Let the three numbers in a G.P. be $a$, $ar$, $ar^2$. By the problem, $\begin{aligned} a+a r+a r^{2}&=42 \\\\ a\left(1+r+r^{2}\right)&=42 \ldots(1)\\\\ a\cdot ar \cdot ar^{2}&=512\\\\ (a r)^{3}&=512\\\\ a r&=8 \\\\ a&=\dfrac{8}{r} \ldots(2) \end{aligned}$ Substituting $a=\dfrac{8}{r}$ in equation (1). $\begin{aligned} \dfrac{8}{r}\left(1+r+r^{2}\right)&=42 \\\\ 4+4 r+4 r^{2}&=21 r \\\\ 4 r^{2}-17 r+4&=0 \\\\ (4 r-1)(r-4)&=0\\\\ \therefore\ r=\frac{1}{4} \text { or } r&=4 \end{aligned}$ When $r=\dfrac{1}{4}$, $a=\dfrac{8}{1/4}=32$ Then three numbers are $32,8,2$. When $r=4$, $a=\dfrac{8}{4}=2$ Then three numbers are $2,8,32$.



23. Insert two geometric means between $2$ and $128$ .





Let the required geometric means be $x$ and $y$, $\therefore 2, x, y, 128$ is a G.P. Let the common ratio be $r$. $\begin{aligned} \therefore\ 2 &=2 \\\\ x &=2 r \\\\ y &=2 r^{2} \\\\ 128 &=2 r^{2}\\\\ \therefore\ r^{3} &=64 \\\\ r &=4 \\\\ \therefore\ x &=8 \\\\ y &=32 \end{aligned}$



24. The ratios of two numbers is $9: 1$. The sum of the arithmetic mean and geometric mean between the two numbers is $96$ , find the two numbers.





Let the two numbers be $9 x$ and $x$. $\begin{aligned} \text { A.M between } 9 x \text { and } x&=\frac{9 x+x}{2}\\\\ &=5 x \\\\ \text { G.M between } 9 x \text { and } x&=\sqrt{9 x^{2}}\\\\ &=3 x \\\\ \text { A.M } + \text { G.M } &=96 \\\\ 5 x+3 x &=96 \\\\ 8 x &=96 \\\\ x &=12 \\\\ 1^{\text {st} } \text {number } &=108 \\\\ 2^{\text {n d} } \text { number } &=12 \end{aligned}$



25. If the arithmetic mean between $x$ and $y$ is $15$ and the geometric mean is $9$ , find $x$ and $y$.





$\begin{aligned} \text{A.M between } x \text{ and } y &=15\\\\ \therefore \ \dfrac{x+y}{2} &=15 \\\\ x+y &=30\ldots(1)\\\\ \text{G.M between } x \text{ and } y = 9\\\\ \therefore \ \sqrt{x y} &=9 \\\\ x y &=81 \\\\ y &=\dfrac{81}{x}\ldots(2)\\\\ \text{Substituting } y=\dfrac{81}{x} &\text{ in equation } (1),\\\\ x+\dfrac{81}{x}&=30 \\\\ x^{2}+81&=30 x \\\\ x^{2}-30 x+81&=0 \\\\ (x-3)(x-27)&=0 \\\\ x=3 \text { or } x&=27 \\\\ \text { When } x=3, y&=\dfrac{81}{3}=27 \\\\ \text { When } x=27, y&=\dfrac{81}{27}=3 \end{aligned}$



26. Show that the products of the corresponding terms of the sequences $a, a r, a r^{2}, \ldots, a r^{n-1}$ and $A, A R, A R^{2}, \ldots, A R^{n-1}$ form a G.P, and find the common ratio.





The first sequence is $a, a r, a r^{2}, \ldots, a r^{n-1}$. The second sequence is $A, A R, A R^{2}, \ldots, A R^{n-1}$. The sequence formed by the products of the corresponding terms of above sequences is $a A, a r A R, a r^{2} A R^{2}, \ldots, a r^{n-1} A R^{n-1}$. $\begin{aligned} &\frac{u_{2}}{u_{1}}=\frac{a r A R}{a A}=r R \\\\ &\frac{u_{3}}{u_{2}}=\frac{a r^{2} A R^{2}}{a r A R}=r R \end{aligned}$ $\therefore$ The ratio of any two successive terms is the same. $\therefore$ The products of the corresponding terms of the sequences $a, a r, a r^{2}, \ldots, a r^{n-1}$ and $A, A R, A R^{2}, \ldots, A R^{n-1}$ form a G.P.



27. Given that $\dfrac{a+b x}{a-b x}=\dfrac{b+c x}{b-c x}=\dfrac{c+d x}{c-d x}$ where $x \neq 0$, prove that $a, b, c, d$ are in G.P.





$\begin{aligned} \text { Let } \dfrac{a+b x}{a-b x}&=\dfrac{b+c x}{b-c x}=\dfrac{c+d x}{c-d x}=k. \\\\ \therefore\ a+b x &=k(a-b x) \\\\ a+b x &=k a-k b x \\\\ k b x+b x &=k a-a \\\\ b x(k+1) &=a(k-1) \\\\ \dfrac{b}{a} &=\dfrac{k-1}{(k+1) x} \\\\ \text { Similarly, } & \\\\ b+c x &=k(b-c x) \\\\ b+c x &=k b-k c x \\\\ k c x+c x &=k b-b \\\\ c x(k+1) &=b(k-1)\\\\ \dfrac{c}{b} & =\dfrac{k-1}{(k+1) x} \\\\ \text { Again, } & \\\\ c+d x & =k(c-d x) \\\\ c+d x & =k c-k d x \\\\ k d x+d x & =k c-c \\\\ d x(k+1) & =c(k-1) \\\\ \dfrac{d}{c} & =\dfrac{k-1}{(k+1) x} \\\\ \therefore\ \dfrac{b}{a}=\dfrac{c}{b} & =\dfrac{d}{c} \\\\ \therefore\ a, b, c, d & \text { is in a G.P.} \end{aligned}$



28. If $a, b, c, d$ are in G.P., prove that $a+b, b+c$ and $c+d$ are also in G.P.





$a, b, c, d$ are in G.P. Let the common ratio be $r$. $\begin{aligned} \therefore a &=a \\\\ b &=a r \\\\ c &=a r^{2} \\\\ d &=a r^{3} \\\\ \dfrac{b+c}{a+b} &=\dfrac{a r+a r^{2}}{a+a r} \\\\ &=\dfrac{r(a+a r)}{a+a r} \\\\ &=r\\\\ \dfrac{c+d}{b+c} &=\dfrac{a r^{2}+a r^{3}}{a r+a r^{2}} \\\\ &=\dfrac{r\left(a r+a r^{2}\right)}{a r+a r^{2}} \\\\ &=r \\\\ \therefore\ \dfrac{b+c}{a+b} &=\dfrac{c+d}{b+c}\\\\ \therefore\ a+b,\ & b+c,\ c+d \text{ are in G.P.} \end{aligned}$



29. If $a, b, c$ are in A.P., show that $2^{a x+1}$, $2^{b x+1}$, $2^{c x+1}$ are in G.P.





$a, b, c$ are in A.P. Let the common difference be $d$. $\begin{aligned} \therefore a &=a \\\\ b &=a+d \\\\ c &=a+2 d \\\\ \dfrac{2^{b x+1}}{2^{a x+1}} &=2^{(b-a) x}=2^{dx}\\\\ \dfrac{2^{c x+1}}{2^{b x+1}} &=2^{(c-b) x}=2^{dx} \\\\ \therefore\ \dfrac{2^{b x+1}}{2^{a x+1}}&=\dfrac{2^{c x+1}}{2^{b x+1}} \\\\ \therefore\ 2^{a x+1},\ & 2^{b x+1},\ 2^{c x+1} \text { are in G.P. } \end{aligned}$



30. The first three terms of an infinite geometric progression are $m-1,6, m+4$, where $m$ is an integer. Show that $m$ satisfies the equation $m^{2}+3 m-40=0$. Hence find the possible values of $m$ and the common ratio of the progression.





\begin{aligned} m-1,\ 6,\ m+4,& \cdots \text { is a GP. } \\\\ \therefore\ \dfrac{6}{m-1} &=\dfrac{m+4}{6} \\\\ m^{2}+3 m-4 &=36 \\\\ \therefore m^{2}+3 m-40 &=0 \\\\ (m+8)(m-5) &=0 \\\\ \therefore\ m=-8 \text { or } m &=5\\\\ \text { When } m &=-8, \\\\ \text { common ratio } &=\dfrac{6}{-8-1} \\\\ &=-\dfrac{2}{3} \\\\ \text { When } m &=5, \\\\ \text { common ratio } &=\dfrac{6}{5-1} \\\\ &=\dfrac{3}{2} \end{aligned}

Exponents and Radicals : Exercise (2.4) - Solution

1. Simplify the following.
   
   (a) $3 \sqrt{5}+7 \sqrt{5}$
   
   
   
   

   $ \begin{aligned} &3\sqrt{5}+7\sqrt{5}\\\\ =&10\sqrt{5}\end{aligned}$

   
   
   (b) $\sqrt{75}-\sqrt{12}$
   
   
   
   

   $\begin{aligned} &\sqrt{{75}}-\sqrt{{12}}\\\\ =&\sqrt{{25\times 3}}-\sqrt{{4\times 3}}\\\\ =&5\sqrt{3}-2\sqrt{3}\\\\ =&3\sqrt{3} \end{aligned}$

   
   
   (c) $3 \cdot 3 \sqrt{3} \cdot 3 \sqrt{27}$
   
   
   
   

   $\begin{aligned} &3\cdot 3\sqrt{3}\cdot 3\sqrt{{27}}\\\\ =&27\cdot \sqrt{3}\cdot \sqrt{{27}}\\\\ =&27\cdot \sqrt{3}\cdot \sqrt{{9\times 3}}\\\\ =&27\cdot \sqrt{3}\cdot 3\sqrt{3}\\\\ =&81\cdot \sqrt{3}\cdot \sqrt{3}\\\\ =&81\cdot 3\\\\ =&243 \end{aligned}$

   
   
   (d) $2 \sqrt{5} \cdot 3 \sqrt{2}$
   
   
   
   

   $\begin{aligned} &2 \sqrt{5} \cdot 3 \sqrt{2}\\\\ =&6\sqrt{10} \end{aligned}$

   
   
   (e) $(4-\sqrt{3})^{2}$
   
   
   
   

   $\begin{aligned} &(4-\sqrt{3})^2\\\\ =&16-2\times 4\times \sqrt{3}+3\\\\ =&19-8\sqrt{3} \end{aligned}$

   
   
   (f) $(\sqrt{3}+2 \sqrt{2})(\sqrt{3}+\sqrt{2})$
   
   
   
   

   $\begin{aligned} &\left( {\sqrt{3}+2\sqrt{2}} \right)\left( {\sqrt{3}+\sqrt{2}} \right)\\\\ =&\sqrt{3}\left( {\sqrt{3}+2\sqrt{2}} \right)+\sqrt{2}\left( {\sqrt{3}+2\sqrt{2}} \right)\\\\ =&3+2\sqrt{6}+\sqrt{6}+4\\\\ =&7+3\sqrt{6} \end{aligned}$

   
   
   (g) $(\sqrt{7}-\sqrt{6})(\sqrt{7}+\sqrt{6})(\sqrt{x}+1)(\sqrt{x}-1)$
   
   
   
   

   $\begin{aligned} &\left( {\sqrt{7}-\sqrt{6}} \right)\left( {\sqrt{7}+\sqrt{6}} \right)\left( {\sqrt{x}+1} \right)\left( {\sqrt{x}-1} \right)\\\\ =&\left( {7-6} \right)\left( {x-1} \right)\\\\ =&x-1 \end{aligned}$

   
   
   (h) $\sqrt{75}-\dfrac{3}{4} \sqrt{48}-5 \sqrt{12}$
   
   
   
   

   $\begin{aligned} &\sqrt{{75}}-\frac{3}{4}\sqrt{{48}}-5\sqrt{{12}}\\\\ =&\sqrt{{25\times 3}}-\frac{3}{4}\sqrt{{16\times 3}}-5\sqrt{{4\times 3}}\\\\ =&5\sqrt{3}-\frac{3}{4}\times 4\sqrt{3}-5\times 2\sqrt{3}\\\\ =&5\sqrt{3}-3\sqrt{3}-10\sqrt{3}\\\\ =&-8\sqrt{3} \end{aligned}$

   
   
   (i) $\sqrt{2 x^{2}}+5 \sqrt{32 x^{2}}-2 \sqrt{98 x^{2}}$
   
   
   
   

   $\begin{aligned} &\sqrt{{2{{x}^{2}}}}+5\sqrt{{32{{x}^{2}}}}-2\sqrt{{98{{x}^{2}}}}\\\\ =&\sqrt{{2{{x}^{2}}}}+5\sqrt{{16\times 2{{x}^{2}}}}-2\sqrt{{49\times 2{{x}^{2}}}}\\\\ =&\left( {\sqrt{2}x} \right)+\left( {5\times 4\sqrt{2}x} \right)-\left( {2\times 7\sqrt{2}x} \right)\\\\ =&\sqrt{2}x+20\sqrt{2}x-14\sqrt{2}x\\\\ =&7\sqrt{2}x \end{aligned}$

   
   
   (j) $\sqrt{20 a^{3}}+a \sqrt{5 a}+\sqrt{80 a^{3}}$
   
   
   
   

   $\begin{aligned} &\sqrt{{20{{a}^{3}}}}+a\sqrt{{5a}}+\sqrt{{80{{a}^{3}}}}\\\\ =&\sqrt{{4\times 5{{a}^{3}}}}+a\sqrt{{5a}}+\sqrt{{16\times 5{{a}^{3}}}}\\\\ =&2a\sqrt{{5a}}+a\sqrt{{5a}}+4a\sqrt{{5a}}\\\\ =&7a\sqrt{{5a}} \end{aligned}$

   
   
   
2. Rationalise the denominators and simplify.
   
   (a) $\dfrac{2}{\sqrt{5}}$
   
   
   
   

   $\begin{aligned} &\dfrac{2}{\sqrt{5}}\\\\ =&\dfrac{2}{\sqrt{5}}\times\dfrac{\sqrt{5}}{\sqrt{5}}\\\\ =&\dfrac{2\sqrt{5}}{5} \end{aligned}$

   
   
   (b) $\dfrac{5}{2+\sqrt{3}}$
   
   
   
   

   $\begin{aligned} &\dfrac{5}{{2+\sqrt{3}}}\\\\ =&\dfrac{{5\left( {2-\sqrt{3}} \right)}}{{\left( {2+\sqrt{3}} \right)\left( {2-\sqrt{3}} \right)}}\\\\ =&\dfrac{{5\left( {2-\sqrt{3}} \right)}}{{4-3}}\\\\ =&5\left( {2-\sqrt{3}} \right) \end{aligned}$

   
   
   (c) $\dfrac{12}{\sqrt{5}-\sqrt{3}}$
   
   
   
   

   $\begin{aligned} &\dfrac{{12}}{{\sqrt{5}-\sqrt{3}}}\\\\ =&\dfrac{{12\left( {\sqrt{5}+\sqrt{3}} \right)}}{{\left( {\sqrt{5}-\sqrt{3}} \right)\left( {\sqrt{5}+\sqrt{3}} \right)}}\\\\ =&\dfrac{{12\left( {\sqrt{5}+\sqrt{3}} \right)}}{{5-3}}\\\\ =&\dfrac{{12\left( {\sqrt{5}+\sqrt{3}} \right)}}{2}\\\\ =&6\left( {\sqrt{5}+\sqrt{3}} \right) \end{aligned}$

   
   
   (d) $\dfrac{\sqrt{2}+1}{2 \sqrt{2}-1}$
   
   
   
   

   $\begin{aligned} &\dfrac{{\sqrt{2}+1}}{{2\sqrt{2}-1}}\\\\ =&\dfrac{{\left( {\sqrt{2}+1} \right)\left( {2\sqrt{2}+1} \right)}}{{\left( {2\sqrt{2}-1} \right)\left( {2\sqrt{2}+1} \right)}}\\\\ =&\dfrac{{\sqrt{2}\left( {2\sqrt{2}+1} \right)+1\left( {2\sqrt{2}+1} \right)}}{{8-1}}\\\\ =&\dfrac{{4+\sqrt{2}+2\sqrt{2}+1}}{7}\\\\ =&\dfrac{{5+3\sqrt{2}}}{7} \end{aligned}$

   
   
   (e) $\dfrac{\sqrt{7}+3 \sqrt{2}}{\sqrt{7}-\sqrt{2}}$
   
   
   
   

   $\begin{aligned} &\dfrac{{\sqrt{7}+3\sqrt{2}}}{{\sqrt{7}-\sqrt{2}}}\\\\ =&\dfrac{{\left( {\sqrt{7}+3\sqrt{2}} \right)\left( {\sqrt{7}+\sqrt{2}} \right)}}{{\left( {\sqrt{7}-\sqrt{2}} \right)\left( {\sqrt{7}+\sqrt{2}} \right)}}\\\\ =&\dfrac{{\sqrt{7}\left( {\sqrt{7}+\sqrt{2}} \right)+3\sqrt{2}\left( {\sqrt{7}+\sqrt{2}} \right)}}{{7-2}}\\\\ =&\dfrac{{7+\sqrt{{14}}+3\sqrt{{14}}+6}}{5}\\\\ =&\dfrac{{13+4\sqrt{{14}}}}{5} \end{aligned}$

   
   
   (f) $\dfrac{\sqrt{17}-\sqrt{11}}{\sqrt{17}+\sqrt{11}}$
   
   
   
   

   $\begin{aligned} &\dfrac{{\sqrt{{17}}-\sqrt{{11}}}}{{\sqrt{{17}}+\sqrt{{11}}}}\\\\ =&\dfrac{{\left( {\sqrt{{17}}-\sqrt{{11}}} \right)\left( {\sqrt{{17}}-\sqrt{{11}}} \right)}}{{\left( {\sqrt{{17}}+\sqrt{{11}}} \right)\left( {\sqrt{{17}}-\sqrt{{11}}} \right)}}\\\\ =&\dfrac{{17-2\sqrt{{17}}\sqrt{{11}}+11}}{{17-11}}\\\\ =&\dfrac{{28-2\sqrt{{187}}}}{6}\\\\ =&\dfrac{{2\left( {14-\sqrt{{187}}} \right)}}{6}\\\\ =&\dfrac{{14-\sqrt{{187}}}}{3} \end{aligned}$

   
   
   (g) $\dfrac{1}{2 \sqrt{2}-\sqrt{3}}$
   
   
   
   

   $\begin{aligned} &\dfrac{1}{{2\sqrt{2}-\sqrt{3}}}\\\\ =&\dfrac{{1\cdot \left( {2\sqrt{2}+\sqrt{3}} \right)}}{{\left( {2\sqrt{2}-\sqrt{3}} \right)\left( {2\sqrt{2}+\sqrt{3}} \right)}}\\\\ =&\dfrac{{2\sqrt{2}+\sqrt{3}}}{{8-3}}\\\\ =&\dfrac{{2\sqrt{2}+\sqrt{3}}}{5} \end{aligned}$

   
   
   (h) $\dfrac{\sqrt{6}+1}{3-\sqrt{5}}$
   
   
   
   

   $\begin{aligned} &\dfrac{{\sqrt{6}+1}}{{3-\sqrt{5}}}\\\\ =&\dfrac{{\left( {\sqrt{6}+1} \right)\left( {3+\sqrt{5}} \right)}}{{\left( {3-\sqrt{5}} \right)\left( {3+\sqrt{5}} \right)}}\\\\ =&\dfrac{{\sqrt{6}\left( {3+\sqrt{5}} \right)+1\left( {3+\sqrt{5}} \right)}}{{9-5}}\\\\ =&\dfrac{{3\sqrt{6}+\sqrt{{30}}+3+\sqrt{5}}}{4}\\\\ =&\dfrac{{3+\sqrt{5}+3\sqrt{6}+\sqrt{{30}}}}{4} \end{aligned}$

   
   
   
3. Write as a single fraction.
   
   (a) $\dfrac{1}{\sqrt{3}+1}+\dfrac{1}{\sqrt{3}-1}$
   
   
   
   

   $\begin{aligned} & \dfrac{1}{\sqrt{3}+1}+\dfrac{1}{\sqrt{3}-1} \\\\ =& \dfrac{1}{\sqrt{3}+1} \times \dfrac{\sqrt{3}-1}{\sqrt{3}-1}+\dfrac{1}{\sqrt{3}-1} \times \dfrac{\sqrt{3}+1}{\sqrt{3}+1} \\\\ =& \dfrac{\sqrt{3}-1}{(\sqrt{3})^{2}-1^{2}}+\dfrac{\sqrt{3}+1}{(\sqrt{3})^{2}-1^{2}} \\\\ =& \dfrac{\sqrt{3}-1}{3-1}+\dfrac{\sqrt{3}+1}{3-1} \\\\ =& \dfrac{\sqrt{3}-1+\sqrt{3}+1}{2} \\\\ =& \dfrac{2 \sqrt{3}}{2} \\\\ =& \sqrt{3} \end{aligned}$

   
   
   (b) $\dfrac{2}{\sqrt{7}+\sqrt{2}}+\dfrac{1}{\sqrt{7}-\sqrt{2}}$
   
   
   
   

   $\begin{aligned} & \dfrac{2}{\sqrt{7}+\sqrt{2}}+\dfrac{1}{\sqrt{7}-\sqrt{2}} \\\\ =& \dfrac{2}{\sqrt{7}+\sqrt{2}} \times \dfrac{\sqrt{7}-\sqrt{2}}{\sqrt{7}-\sqrt{2}}+\dfrac{1}{\sqrt{7}-\sqrt{2}} \times \dfrac{\sqrt{7}+\sqrt{2}}{\sqrt{7}+\sqrt{2}} \\\\ =& \dfrac{2(\sqrt{7}-\sqrt{2})}{(\sqrt{7})^{2}-(\sqrt{2})^{2}}+\dfrac{\sqrt{7}+\sqrt{2}}{(\sqrt{7})^{2}-(\sqrt{2})^{2}} \\\\ =& \dfrac{2(\sqrt{7}-\sqrt{2})}{7-2}+\dfrac{\sqrt{7}+\sqrt{2}}{7-2} \\\\ =& \dfrac{2 \sqrt{7}-2 \sqrt{2}+\sqrt{7}+\sqrt{2}}{5} \\\\ =& \dfrac{3 \sqrt{7}-\sqrt{2}}{5} \end{aligned}$

   
   
   (c) $\dfrac{1}{3+\sqrt{3}}+\dfrac{1}{\sqrt{3}-3}+\dfrac{1}{\sqrt{3}}$
   
   
   
   

   $\begin{aligned} & \dfrac{1}{3+\sqrt{3}}+\dfrac{1}{\sqrt{3}-3}+\dfrac{1}{\sqrt{3}} \\\\ =& \dfrac{3-3 \sqrt{3}}{\sqrt{3}(3+\sqrt{3})(\sqrt{3}-3)}+\dfrac{3 \sqrt{3}+3}{\sqrt{3}(3+\sqrt{3})(\sqrt{3}-3)}+\dfrac{-6}{\sqrt{3}(3+\sqrt{3})(\sqrt{3}-3)} \\\\ =& \dfrac{3-3 \sqrt{3}+3 \sqrt{3}+3-6}{\sqrt{3}(3+\sqrt{3})(\sqrt{3}-3)} \\\\ =& \dfrac{0}{\sqrt{3}(3+\sqrt{3})(\sqrt{3}-3)} \\\\ =& 0 \end{aligned}$

   
   
   (d) $\dfrac{7+\sqrt{5}}{7-\sqrt{5}}+\dfrac{\sqrt{11}-3}{\sqrt{11}+3}$
   
   
   
   

   $\begin{aligned} & \dfrac{7+\sqrt{5}}{7-\sqrt{5}}+\dfrac{\sqrt{11}-3}{\sqrt{11}+3} \\\\ =& \dfrac{(7+\sqrt{5})(7+\sqrt{5})}{(7-\sqrt{5})(7+\sqrt{5})}+\dfrac{(\sqrt{11}-3)(\sqrt{11}-3)}{(\sqrt{11}+3)(\sqrt{11}-3)} \\\\ =& \dfrac{54+14 \sqrt{5}}{49-5}+\dfrac{20-6 \sqrt{11}}{11-9} \\\\ =& \dfrac{54+14 \sqrt{5}}{44}+\dfrac{20-6 \sqrt{11}}{2} \\\\ =& \dfrac{27+7 \sqrt{5}}{22}+10-3 \sqrt{11} \\\\ =& \dfrac{27+7 \sqrt{5}+220-66 \sqrt{11}}{22} \\\\ =& \dfrac{247+7 \sqrt{5}-66 \sqrt{11}}{22} \end{aligned}$

   
   
   (e) $\dfrac{3+2 \sqrt{2}}{(\sqrt{3}-1)^{2}}$
   
   
   
   

   $\begin{aligned} & \dfrac{3+2 \sqrt{2}}{(\sqrt{3}-1)^{2}} \\\\ =& \dfrac{3+2 \sqrt{2}}{(\sqrt{3})^{2}-2 \sqrt{3+1}} \\\\ =& \dfrac{3+2 \sqrt{2}}{4-2 \sqrt{3}} \times \dfrac{4+2 \sqrt{3}}{4+2 \sqrt{3}} \\\\ =& \dfrac{(3+2 \sqrt{2})(4+2 \sqrt{3})}{4^{2}-(2 \sqrt{3})^{2}} \\\\ =& \dfrac{12+6 \sqrt{3}+8 \sqrt{2}+4 \sqrt{6}}{16-12} \\\\ =& \dfrac{12+6 \sqrt{3}+8 \sqrt{2}+4 \sqrt{6}}{4} \\\\ =& \dfrac{6+3 \sqrt{3}+4 \sqrt{2}+2 \sqrt{6}}{2} \end{aligned}$

   
   
   (f) $\sqrt{\dfrac{x+1}{x-1}}+\sqrt{\dfrac{x-1}{x+1}}-\sqrt{\dfrac{1}{x^{2}-1}}$
   
   
   
   

   $\begin{aligned} & \sqrt{\dfrac{x+1}{x-1}}+\sqrt{\dfrac{x-1}{x+1}}-\sqrt{\dfrac{1}{x^{2}-1}} \\\\ =& \dfrac{\sqrt{x+1}}{\sqrt{x-1}}+\dfrac{\sqrt{x-1}}{\sqrt{x+1}}-\dfrac{1}{\sqrt{x^{2}-1}} \\\\ =&\left(\dfrac{\sqrt{x+1}}{\sqrt{x-1}} \times \dfrac{\sqrt{x-1}}{\sqrt{x-1}}\right)+\left(\dfrac{\sqrt{x-1}}{\sqrt{x+1}} \times \dfrac{\sqrt{x+1}}{\sqrt{x+1}}\right)-\left(\dfrac{1}{\sqrt{x^{2}-1}} \times \dfrac{\sqrt{x^{2}-1}}{\sqrt{x^{2}-1}}\right) \\\\ =& \dfrac{\sqrt{x^{2}-1}}{x-1}+\dfrac{\sqrt{x^{2}-1}}{x+1}-\dfrac{\sqrt{x^{2}-1}}{x^{2}-1} \\\\ =& \dfrac{(x+1) \sqrt{x^{2}-1}}{(x-1)(x+1)}+\dfrac{(x-1) \sqrt{x^{2}-1}}{(x+1)(x-1)}-\dfrac{\sqrt{x^{2}-1}}{x^{2}-1} \\\\ =&\left(\dfrac{x+1+x-1-1}{x^{2}-1}\right) \sqrt{x^{2}-1} \\\\ =& \dfrac{(2 x+1) \sqrt{x^{2}-1}}{x^{2}-1} \end{aligned}$

   
   
   (g) $\sqrt{\dfrac{\sqrt[5]{32}+\sqrt{4}}{2^{-2}-2^{-3}}}$
   
   
   
   

   $\begin{aligned} & \sqrt{\dfrac{\sqrt[5]{32}+\sqrt{4}}{2^{-2}-2^{-3}}} \\\\ =& \sqrt{\dfrac{\sqrt[5]{2^{5}}+\sqrt{4}}{\dfrac{1}{2^{2}}-\dfrac{1}{2^{3}}}} \\\\ =& \sqrt{\dfrac{2+2}{\dfrac{1}{4}-\dfrac{1}{8}}} \\\\ =& \sqrt{\dfrac{4}{1 / 8}} \\\\ =& \sqrt{4 \times 8} \\\\ =& \sqrt{32} \\\\ =& 4 \sqrt{2} \end{aligned}$