Arithmetic Progression

Arithmetic Progression (A . P)

In a sequence the difference of any two consecutive terms is constant, then the sequence is called an arithmetic progression. That difference is called the common difference and is denoted by d.



Sequence တစ္ခုတြင္ ကပ္လွ်က္ရွိေသာ မည့္သည့္ကိန္းႏွစ္ခုမဆို ျခားနားျခင္း(ႏႈတ္ျခင္း) သည္ ကိန္းေသျဖစ္ လွ်င္ ၎ sequence ကို arithmetic progression ဟုေခၚသည္။



If u₁ , u₂ , u₃ , u₄ , - - -, u_n-1 , u_n is an A . P,then u₂ -u₁ = u₃ -u₂= u₄ -u₃ = - - - = u_n -u_n-1 = constant.

u_n -u_n-1 = d.

u_n = u_n-1 + d

Generally the first term (u₁₎ of a sequence is denoted by a.

Therefore ....

u₁ = a

u_n = u_n-1 + d

u₂ = u₁ + d = a + d

u₃ = u₂ + d = 2a + d

u₄ = u₃ + d = 3a + d

From above expression the n^th term of an arithmetic progression can be expressed as

un = a + (n-1)d

where ...

u_n = the n^th term

a = the first term

d = the common difference

n = number of term

Sequences and Series

Sequence



ေအာက္မွာေပးထားတဲ့ ကိန္းတန္းေလးကို ေလ့လာၾကည့္ရေအာင္။



1,4,9,16,25,...



အခုကိန္းတန္းမွာပါ၀င္တဲ့ ကိန္းလံုးေတြဟာ ေရးခ်င္သလို ေရးထားျခင္း (random) မဟုတ္ပါဘူး။ ကိန္းလံုးေတြ ေျပာင္းလဲမႈမွာ စနစ္တစ္ခု (တနည္းေျပာရရင္ function တစ္ခု) အရ ေျပာင္းလဲသြားတာပါ။ ဒါကို functional notation နဲ႔ေျပာရမယ္ဆိုရင္...



f(1) = 1 = 1²

f(2) = 4 = 2²

f(3) = 9 = 3²

f(4) = 16 = 4²

f(5) = 25 = 5² လို႔ဆိုႏိုင္တာေပါ။့



ဒါကိုၾကည့္ျခင္းအားျဖင့္ function ရဲ့ Domain ဟာ {1,2,3,4,5,...n}=the set of natural numbers ဆိုရပါမယ္။ ဒါဆိုရင္ အထက္ပါေျပာင္းလဲမႈကို ၾကည့္ရံုနဲ႔ function ရဲ့ general formula ကို အလြယ္တကူ ေျပာႏိုင္ပါၿပီ။

f(n) = n² ေပါ့...။



ဒါေၾကာင့္ sequence ဆိုတာဟာ special function လို႔ဆိုႏိုင္ၿပီး သူရဲ့ domain ကေတာ့ အၿမဲတမ္း သဘာ၀ကိန္း မ်ား ပါ၀င္ေသာအစု (the set of natural numbers) ျဖစ္ပါတယ္။ image ေတြကိုေတာ့ ဒီေနရာမွာ terms လို႔ ေျပာင္းလဲေခၚပါမယ္။ အေခၚအေ၀ၚ ေျပာင္းလဲမႈနဲ႔အတူ အသံုးျပဳမယ့္ symbols ေတြကိုလည္း ေျပာင္းလည္း သတ္မွတ္ပါတယ္။ အထက္မွာေပးထားတဲ့ ကိန္းလံုးေတြကို အခုလိုေခၚေ၀ၚ သတ္မွတ္ပါမယ္။

first term	=	u1	=	1
second term	=	u2	=	4
third term	=	u3	=	9
fourth term	=	u4	=	16
fifth term	=	u5	=	25
- - - - - - - -	- -	- -	- -	- -
nth term	=	un	=	n2

ဒီေနရာမွာ n^th term ကို general term သို႔မဟုတ္ general formula လို႔ဆိုႏိုင္ပါတယ္။ ဒါေၾကာင့္ sequence ကို အခုလို definition ဖြင့္ဆိုႏိုင္ပါတယ္။



Sequence

A sequence is a function whose domain is either the set of all or part of natural numbers. The values (images) of function are called terms .



Example 1

Find the first four terms of the sequence defined by u_n = 3n - 5.



Solution

u_n = 3n - 5

u₁ = 3(1) - 5 =-2

u₂ = 3(2) - 5 = 1

u₃ = 3(3) - 5 = 4

u₄ = 3(4) - 5 = 7

Therefore the fist four terms are -2, 1, 4, 7.



Exercises

Find the first four terms of the sequence defined by

(a) un = 2n + 3

(b) un = 3n2 - 2

(c) un = 4n2+ 3n - 5

Example2

Which term of the sequence defined by u_n = 4n - 23 is 25?



Solution

u_n = 4n - 23

Let the n^th term be 25.

Therefore u_n = 25

4n - 23 = 25

4n = 48

n = 12

Therefore the 12^th term is 25.