How many terms of the Arithmetic Progression 45, 39, 33, ... must be taken so that their sum is 180 ? Explain the double answer.

### Answer

S_{n} = 180

n/2 [90 + (n - 1)(-6)] = 180

90n – 6n^{2} + 6n = 360

⇒ 6n^{2} – 96n + 360 = 0

⇒ 6[(n – 6)(n – 10)] = 0

⇒ n = 6, n = 10

Sum of a_{7}, a_{8}, a_{9}, a_{10} = 0

∴ n = 6 or n = 10