How many terms of the ap 9,17,25, ... must be taken to get a sum of 450

The AP is 9, 17, 25, ...

a (first term) = 9

d (common difference) = 17-9 = 8

Let n be the required number of terms.

Sn (sum of the n terms) = 450

We know that Sn = n/2 (2a + (n-1) d)

450 = n/2 (18 + (n-1) 8)

900 = n (18 + 8n - 8)

900 = 8n2 + 10n

8n2 + 10n - 900 = 0

8n2 + 80n - 90n - 900 = 0

8n (n + 10) - 90 (n - 10) = 0

(n-10) (8n-90) = 0

n = 10 or 11.25

But n has to be a natural, since number of terms is always a natural number.

Thus, the required number of terms is 10.