[vc_row][vc_column][vc_column_text]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?[latexpage][/vc_column_text][/vc_column][/vc_row][vc_row][vc_column][lvca_accordion][lvca_panel panel_title="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 $n-2$[/lvca_panel][/lvca_accordion][vc_column_text]
Problem Page Coordinator: Stephen Lee CMath MIMA – Mathematics in Education and Industry
Acknowledgement: The IMA are indebted to MEI for sourcing and supplying Mathematics Today with these well-known puzzles.
First published in Mathematics Today (December 2015)
