Means Between Numbers

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If three arithmetic means are inserted between the numbers 4 and 28 to form an arithmetic progression, what is the value of the middle inserted mean?

RankGuru · Accepted Answer

Inserting arithmetic means between two numbers creates an arithmetic progression in which the original numbers are the endpoints, a classic JEE Advanced construction. With three means inserted, the sequence has five terms total: 4, A_1, A_2, A_3, 28. The common difference is d = (28 - 4)/(5 - 1) = 24/4 = 6. The three means are 10, 16, 22, so the middle mean A_2 equals 16. Option 12 results from using four intervals incorrectly as three. Option 20 takes the third mean by miscounting. Option 14 averages the wrong pair of neighbours. Hence the middle mean is 16. The general principle is that inserting m means always produces m + 1 equal gaps spanning the endpoints, so the common difference scales inversely with the number of inserted terms, a fact JEE Advanced reuses when comparing different insertion counts. Plausibility check: the middle of three equally spaced means must equal the arithmetic mean of the endpoints, and (4 + 28)/2 = 16, which independently confirms the result through the symmetry of an AP about its centre.

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