MEI Maths Item of the Month
June 2014
APGP
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?
Solution
The 1st, 2nd and 6th terms of an AP are a, a  d , a  5d .
a  d a  5d
These three terms forming a GP gives:
.

a
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 ration 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 ration of n  2 .
1 of 1
TB v1.1 22/10/2015
© MEI