Mathematics

Mathematics

Problem

The sum of the first n terms of an arithmetic progression with first term 8 is 360. The common difference d is required to be an integer. (i) Establish an equation linking n and d with the given sum. (ii) Determine all pairs of positive integers (n, d) that satisfy the equation. (iii) For each pair obtained in part (ii), write the corresponding progression up to its nth term.
Intuitive explanation

An arithmetic progression's sum depends on both how many terms you include and how much they grow—these two constraints are locked together. Your job is to use the sum formula Sn=n2(2a+(n−1)d)S_n = \frac{n}{2}(2a + (n-1)d) to create one equation in two unknowns, then hunt for all integer solutions by recognizing that nn must divide evenly into the rearranged equation. This is a number-theory puzzle hiding inside algebra: once you isolate dd, you'll see which values of nn actually yield integer values for dd.

Intuitive explanation

An arithmetic progression's sum depends on both how many terms you include and how much they grow—these two constraints are locked together. Your job is to use the sum formula Sn=n2(2a+(n−1)d)S_n = \frac{n}{2}(2a + (n-1)d) to create one equation in two unknowns, then hunt for all integer solutions by recognizing that nn must divide evenly into the rearranged equation. This is a number-theory puzzle hiding inside algebra: once you isolate dd, you'll see which values of nn actually yield integer values for dd.

Browse the problem library