**Q) **How many terms of the arithmetic progression 45, 39, 33, …….. must be taken so that their sum is 180? Explain the double answer.

**Ans:**

In AP of 45,39, 33, …….

a = 45, d = – 6,

Sum of n terms of AP, S_{n} = (2a + (n-1)d)

180 = (2 x 45 + (n-1)(-6))

360 = n (90 – 6 (n-1))

6n^{2} – 96n + 360 = 0

n^{2} – 16n + 60 = 0

(n – 10) (n – 6) = 0

Hence, for n = 6 and n = 10 or we can say that for sum of AP is 180 for first 6 terms and also for first 10 terms.

**Reason for 2 values: **We get two values of ‘n’ because 9^{th} and 10^{th} term values are negative and cancel out with values of 7^{th} and 8^{th} terms. Hence, sum of first 6 terms and sum of 10 terms are 180.