Question

How many terms of the AP 18 16, 14, 12, …. are needed to give the sum 78? Explain the double answer.

Answer 1

To Find: Number of terms required to make the sum 78.

Here a = 18, d = - 2

Let n be the number of terms required to make the sum 78.

⇒ 78 × 2 = 36n - 2n² + 2n

⇒ n² - 19n + 78 = 0

⇒ n² - 6n - 13n + 78 = 0

⇒ n(n - 6) - 13(n - 6) = 0

⇒ (n - 13)(n - 6) = 0

Either n = 13 or n = 6

Explanation: Since the given AP is a decreasing progression where a_{n - 1} > a_n,it is bound to have negative values in the series. S_n is maximum for n = 9 or n = 10 since T₁₀ is 0(S₁₀ = S₉ = S_max = 90). The sum of 78 can be attained by either adding 6 terms or 13 terms so that negative terms from T₁₁ onward decrease the maximum sum to 78.