Mathematics

Mathematics

Problem

Between the numbers −4-4 and 1616 five arithmetic means are to be inserted so that all the seven numbers (including the given ones) form an arithmetic progression. (i) Find the common difference of this progression. (ii) List all five arithmetic means in ascending order. (iii) Calculate the sum of only the five means obtained in part (ii).
Intuitive explanation

An arithmetic progression is a sequence where consecutive terms differ by a constant amount—the common difference dd. When you insert means between two given numbers, you're filling in the missing terms of a longer AP. The key is to use the formula for the nn-th term, an=a1+(n−1)da_n = a_1 + (n-1)d, to find dd from the endpoints, then generate all intermediate terms systematically.

Intuitive explanation

An arithmetic progression is a sequence where consecutive terms differ by a constant amount—the common difference dd. When you insert means between two given numbers, you're filling in the missing terms of a longer AP. The key is to use the formula for the nn-th term, an=a1+(n−1)da_n = a_1 + (n-1)d, to find dd from the endpoints, then generate all intermediate terms systematically.

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