ARITHMETIC PROGRESSION :IMPORTANT FORMULAS ,SHORTCUTS AND TRICKS

Arithmetic Progressions

An arithmetic progression(AP) is a sequence of numbers in which each term is obtained by adding a fixed number to the preceding term .

The general form of an arithmetic progression (AP) is :

a ,a+d, a+2d,a+3d……..

Where a=first term and

d= common difference

Example : Write the 1st term and the common difference.
1) 3,1,-1,-3………
First term a=3
Common difference d= a2- a1= -2

2)Which of the following sequence form an AP?
i. 4,10,16,22….
ii. -2,2,-2,2,-2
Ans :i) Given term ; 4,10,16,22….
a2- a1=10-4=6
a3- a2=16-10=6
a4- a3=22-16=6
The given list of numbers form an AP with common difference d=6

Ans :ii)Given term -2, 2,-2,2,-2
a2- a1=2- (-2)=4
a3- a2=-2-2 =-4
a2- a1 ≠ a3- a2,
Hence the given numbers does not forms an AP .

Arithmetic Progression Important Formulas

n^th term of an AP

In an Arithemetic Progression if a is the first term and d is the common difference then nth term is given by the formula

a_n =a+ (n-1)d

Example: Find the 10th term of the AP: 2,7,12,……

Ans: Here a=2 , d= 5
n=10
a_n =a+ (n-1)d
a₁₀ =2+ (10-1)×5 =47
10th term of given AP is 47

Example:Which term of the Aritmetic Progression : 21, 18,15 ,….. is - 81

Ans) a= 21 , d= -3
a_n= -81, the n=?
a_n =a+ (n-1)d , -81= a+ (n-1)d
-81 = 21+(n-1) × -3 ,
(n-1) × -3 =-81 -21= - 102
(n-1) = (- 102 / -3)= 34
n=34+1=35

Exmple :Determine AP whose 3rd term is 5 and 7th term is 9?

a_n =a+ (n-1)d
a₃ =a+ (3-1)d = 5 …..(1)
a₇ =a+ (7-1)d = 9…….(2)
Solving the two equations we get; a=3 and d= 1
The required AP is 3,4,5,6,7….

SUM OF n TERMS OF AN A.P

The sum of first n terms of an A.P= n/2[2a + (n−1) d]
If l is the last term of a finite sequence ,then sum of all terms of an AP is given by
Sn= n/2[a+l]

Arithmetic mean

If a,b,c are in AP, then b is called arithmetic mean of a and c

Then b = (a+c)/2

Example :Find the sum of the first 22 terms of the A.P; 8,3,-2……?

Here a=8 , d=3-8=-2-3 = -5 Sum = n/2[2a + (n−1) d] = 22/2 (2× 8 +(22-1) × -5) =979