How many terms of the series 15 + 12 + 9 + ... must be taken to make 15? Explain the double answer.

Here, a = 15, d = 12 – 15 = -3 and S_n = 15

We know that,

⇒ 30 = n[33 – 3n ]

⇒ 3n² – 33n + 30 = 0

⇒ 3n² – 30n – 3n + 30 = 0

⇒ 3n(n – 10) -3(n – 10) = 0

⇒ (n – 10)(3n – 3) = 0

⇒ n – 10 = 0 or 3n – 3 = 0

⇒ n = 10 or n = 1

The number of terms can be 1 or 10.

Here, the common difference is negative.

∴ The AP starts from a positive term, and its terms are decreasing.

∴ All the terms after 6^th term are negative.

We get a double answer because these positive terms from 2^nd to 5^th term when added to negative terms from 7^th to 10^th term, they cancel out each other and the sum remains same.