How many terms of the AP : 9, 17, 25, . . . must be taken to give a sum of 636?

Given that, a = 9, d = 8 and S_n = 636

Let no of terms be N.

Now

We know:

⇒ 636

⇒ 636= (18 + 8N -8)

⇒ 636 = N(9 +4N -4)

⇒ 636 = 9N + 4N² - 4N 

⇒ 4N² + 5N -636 =0
Now to factorize the above quadratic equation, we need to make the product ( 636 . 4 = 2544) and difference should be 5
⇒ 4N² + 53N - 48N -636 = 0

⇒ N(4N + 53) - 12(4N + 53) = 0
⇒ (N - 12)(4N + 53) = 0

Now, N = - 53/4 and N = 12

Taking the integral value and rejecting the fractional value,
we have n = 12