Find the number of terms:

How many terms of the arithmetic progression 9, 17, 25, … be taken so that their sum is 636?

Here first term a = 9 and common difference d = 17 – 9 = 8.
Also, S_n = 636
Since the sum of n terms is
S_{n =}

Where,

a = First term of AP
d = Common difference of AP
and no of terms is ‘n’

636 = n[ 9 + (n - 1)4]
636 = 9n + 4n(n - 1)
636 = 9n + 4n² – 4n
636 = 4n² + 5n
4n² + 5n – 636 = 0
⇒ n = 12, n = -13.25 (which is not applicable ∵ n can neither be negative nor be in decimal)
∴ n = 12

Hence, 12 terms of the arithmetic progression must be taken to get sum as 636.