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

When you try this calculation on your own, use this *arithmetic progression calculator* to verify your answers.

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