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Sum of first 10 terms is equal to sum of first 12 terms. Sum of first 12 terms = Sum of first 10 terms + 11th term + 12th term 11th term + 12thterm = 0 Let 11thterm = k, common ratio = r 12thterm will be = kr k + kr = 0 k (1 + r) = 0 r = -1 as k cannot be zero Common ratio = -1 Now, sum of 15 terms = a(r - 1)/(r - 1) = 31 ⇒ a(-1 -1)/(-1 -1) = 31 ⇒ a = 31 ∴ Third term = ar = 31 × (-1) = 31
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