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What will be the value of p if log516, log5(3p – 4), log5(3p + 97 / 16) are in arithmetic progression?

What will be the value of p if log516, log5(3p – 4), log5(3p + 97 / 16) are in arithmetic progression? Correct Answer 3

Given, log516, log5(3p – 4), log5(3p + 97 / 16) are in arithmetic progression. ➩ 2 log5 (3p – 4) = log516 + log5(3p + 97 / 16) ➩ log5 (3p – 4)2 = log5 ➩ (3p – 4)2 = 16(3p + 97 / 16) Let 3p = q ➩ (q – 4)2 = 16(q + 97 / 16) ➩ q2 – 8q + 16 = 16q + 97 ➩ q2 – 24q – 81 = 0 ➩ q2 – 27q + 3q – 81 = 0 ➩ q (q – 27) + 3(q – 27) = 0 ➩ (q + 3) (q – 27) = 0 ➩ q = -3, 27 As 3p is a positive number therefore, rejecting – 3. Thus, 3p = 27 ➩ 3p = 33 ⟹ p = 3

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