# HOW TO FIND THE SUM OF SERIES IN AP WITH GIVEN INFORMATION

How to Find the Sum of Series in AP with Given Information ?

To find the sum of the series, we use the formula given below.

Sn  =  (n/2) [a + l] (or)

Sn  =  (n/2) [2a + (n - 1)d]

To find the first term, common difference and number of term, we use the formula given below.

an  =  a + (n - 1)d

## How to Find the Sum of Series in AP with Given Information - Examples

Question 1 :

The first and last term of an AP are 17 and 350 respectively.If the common difference is 9, how many terms are there and what is their sum?

Solution :

a = 17       l = 350   d = 9

Sn  =  (n/2) [2a + (n-1)d]

Sn  =  (n/2) [2(17)+(n-1)9]

=  (n/2) [34 + 9 n - 9]

=  (n/2) [25 + 9 n]  ----(1)

Sn  =  (n/2)[a + l]

=  (n/2)[17 + 350] ------(2)

(1)  =  (2)

(n/2) [25 + 9 n]  =  (n/2)[17 + 350]

25 + 9 n = 17 + 350

9n  =  367 - 25

9n  =  342

n = 342/9

n  = 38

38 terms are needed.

S38 = (38/2)[17 + 350]

=  19 

=  6973

Question 2 :

The sum of first 22 terms of an AP in which d = 7and 22nd term is 149.

Solution :

n = 22  d = 7

a22  =  a + 21 d

=  a + 21 (7)

149 = a + 147

149 -147 = a

a = 2

Sn  =  (n/2) [2a + (n-1)d]

=  (22/2) [2(2) + (22-1)7]

=  11 [4 + 21(7)]

=  11 [4 + 147]

=  11 

=  1661

Question 3 :

The sum of first 51 terms of an AP whose second and third terms are 14 and 18 respectively.

Solution :

a₂ = a + 1d

14 = a + d

a + d = 14 ---(1)

a₃ = a + 2d

18 = a + 2d

a + 2 d = 18 ------(2)

(1) - (2)

a + d = 14

a + 2 d = 18

(-)   (-)   (-)

--------------

- d = -4

d = 4

By applying the value of d = 4 in the (1), we get

a + 4  =  14

a = 14 - 4

a = 10

Sn = (n/2) [2a + (n - 1) d]

n = 51

Sn  =  (51/2)[2(10) + (51-1) 4]

=  (51/2)[20 + 50(4)]

=  (51/2) [20 + 200]

=  (51/2) 

=  51 (110)

S51  =  5610 After having gone through the stuff given above, we hope that the students would have understood, how to find the sum of series in ap with given information.

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