Mathematics

Problem

Three arithmetic means are inserted between a number

x

and

35

. The five numbers obtained in this way have a combined sum of

151

. (i) Express the three means in terms of

x

and the common difference. (ii) Using the information about the sum, form an equation to determine

x

. (iii) Hence find the three arithmetic means.

Intuitive explanation

An arithmetic sequence is defined by a constant step between consecutive terms. When you insert three means between $x$ and $35$ , you're creating a 5-term sequence where the common difference $d$ links all terms together. The key insight is that the sum of an arithmetic sequence depends only on the first term, last term, and how many terms there are—so you can use the given sum of $151$ to find both $x$ and $d$ without solving for them separately first.

Intuitive explanation

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