This below worksheet help you to understand how to calculate Arithmetic Progression. A Progression is the succession of numbers formed and arranged in a definite order according to the certain definite rule. Whereas If each term of a progression differ from its preceding term by a constant, then such a progression is called an arithmetic progression and the constant different is called the common difference of the AP denoted by d. The general expression of A.P is a, (a+d), (a+2d), (a+3d), ....

Arithmetic Progression Formula The nth term of this A.P. is given by Tn = a + (n-1)d
The sum of n terms of this A.P.
Sn = n/2[ 2a + (n - 1)d] = (n/2)[first term + last term]

Arithmetic Progression Example How many numbers between 8 and 121 are divisible by 11
The required numbers are 11, 22, 33, 44, 55, ....., 110, 121
Here a= 11 & d = 22 - 11 = 11 Tn = 121
Tn = a + (n - 1)d
121 = 11 + (n - 1)11
n - 1 = (121 - 11) / 11
= 110 / 11
= 10
n = 10 + 1
n = 11

1) Find the sum of all odd numbers upto 85.
Solution:
Given numbers are 1, 3, 5, 7, 9, ...... ,85
Here a = 1 & d = 3 - 1 = 2.
Tn = a + (n-1)d
85 = 1 + (n - 1)2
n - 1 = (85 - 1) / 2
n - 1 = 42
n = 43.
Sum = (n / 2)(first term + last term)
= 43 / 2 (1 + 85)
= 1849

Hence the required Sum is 1849

2) Find the sum of all 2 digit numbers divisible by 4.
Solution:
All the 2 digits numbers divisible by 4 are,
12, 16, 20, ...... ,96
Here a = 12 & d = 16 - 12 = 4.
Tn = a + (n-1)d
96 = 12 + (n - 1)4
n - 1 = (96 - 12)/4
= 84/4
= 21
n = 22
Sum = (n / 2)(first term + last term)
= (22 /2) x (12 + 96)
= 11 x 108
= 1188

Hence the required sum is 1188.

Notes:
Some of the Important Results to Remember:
Arithmetic Progression for Consecutive Number (1 + 2 + 3 + ..... + n) = (n x (n + 1)) / 2

Arithmetic Progression for Consecutive Square (1^{2}+2^{2}+3^{2}+ ..... +n^{2}) = (n x (n + 1)x(2n + 1)) / 6

Arithmetic Progression for Consecutive Cubes (1^{3}+2^{3}+3^{3}+ ..... +n^{3}) = (n^{2} x (n + 1)^{2}) / 4