How many terms of the A.P. 24, 21, 18, ... must be taken so that the sum is 78 ? Explain the double answer.

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KnowledgeBoat · Accepted Answer

Let numbers of terms be n. Given, a = 24, d = 21 - 24 = -3 and S_n = 78. By formula S_n = ⇒ 78 = ⇒ 78 × 2 = n[48 - 3n + 3] ⇒ 156 = n[51 - 3n] ⇒ 156 = 51n - 3n² ⇒ 3n² - 51n + 156 = 0 ⇒ 3n² - 12n - 39n + 156 = 0 ⇒ 3n(n - 4) - 39(n - 4) = 0 ⇒ (3n - 39)(n - 4) = 0 ⇒ 3n - 39 = 0 or n - 4 = 0 ⇒ 3n = 39 or n = 4 ⇒ n = 13 or n = 4. Let's check sum of 4 terms ⇒ a₄ = a₃ + d ⇒ a₄ = 18 + (-3) = 15 ∴ S₄ = 24 + 21 + 18 + 15 = 78. By formula, a_n = a + (n - 1)d ⇒ a₅ = 24 + (5 - 1)(-3) ⇒ a₅ = 24 + 4(-3) ⇒ a₅ = 24 - 12 = 12. a₆ = a₅ + d = 12 + (-3) = 9 a₇ = a₆ + d = 9 + (-3) = 6 a₈ = a₇ + d = 6 + (-3) = 3 a₉ = a₈ + d = 3 + (-3) = 0 a₁₀ = a₉ + d = 0 + (-3) = -3 a₁₁ = a₁₀ + d = -3 + (-3) = -6 a₁₂ = a₁₁ + d = -6 + (-3) = -9 a₁₃ = a₁₂ + d = -9 + (-3) = -12. Taking sum of from 5th term to 13th, 12 + 9 + 6 + 3 + 0 - 3 - 6 - 9 - 12 = 0 Since, the sum from 5tn term to 13th is zero hence, it will not add a value. ∴ n = 13 and 4. Hence, the number of terms that can be taken for sum to be 78 can be 4 or 13.

Mathematics

How many terms of the A.P. 24, 21, 18, … must be taken so that the sum is 78 ? Explain the double answer.

AP GP

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