Common ratio and progressions puzzle

An arithmetic progression (AP) is a sequence of numbers with a common difference between them, e.g. 3, 7, 11, 15, …

A geometric progression (GP) is a sequence of numbers with a common ratio between them, e.g. 2, 6, 18, 54, …

If the 1st, 2nd and 6th terms of an AP form a GP what is the common ratio?

If the 1st, 2nd and nth terms of an AP form a GP what is the common ratio?

Reveal Solution

The 1st, 2nd, and 6th terms of an AP are $a, a + d, a +5d$

These three terms forming a GP gives: $\frac{a + d}{a}=\frac{a + 5d}{a+d}$

Rearranging this gives:

$a^2 +2ad +d^2 = a^2 +5ad$

$2ad +d^2 = 5ad$

$d^2 = 3ad$

$d = 3a$

(or $d= 0$ which isn’t very interesting)

Therefore the GP has first two terms $a, 4a$ and hence the common ratio is 4.

Similarly, the 1st, 2nd and nth terms of an AP are $a, a+d, a+(n-1)d$

This gives $d=(n-3)a$ and hence a common ratio of $n-2$

Published 1st December 2015