Terms in Geometric Progressions

Sometimes we are given just enough information about a geometric progression to find its common ratio. Recall that the $n$nth term of a GP is given by:

$t_n=ar^{n-1}$tn​=arn−1

Note that there are two variables in the formula – the first term $a$a and the common ratio $r$r. It is often the case in mathematics that in order to find the value of these parameters we need to be given two separate pieces of information.

Carefully consider the following example:

Suppose that for a certain geometric progression we are told that its third term is $63$63 and its fifth term is $567$567. Hence we know that:

$63=ar^2$63=ar2

$567=ar^4$567=ar4

Here we have two equations and two unknowns, so we should be able to find the first term and common ratio of the sequence. We do this by division, so that:

$\frac{567}{63}=\frac{ar^4}{ar^2}$56763​=ar4ar2​

By cancelling factors this equation becomes $9=r^2$9=r2 and so $r=\pm3$r=±3 . This means there are two possible geometric progressions to consider.

Firstly, if $r=3$r=3, then since  we have that $63=9a$63=9a and so $a=7$a=7. So the first five terms of the sequence become $7,21,63,189$7,21,63,189 and $567$567. Secondly, if $r=-3$r=−3 then $a$a is again $7$7, but every second term changes sign so that the first five terms become $7,-21,63,-189$7,−21,63,−189 and $567$567. Note that both of these sequences have the third and fifth term $63$63 and $567$567 respectively as required by the original information given in the question.

Worked Examples

QUESTION 1

QUESTION 2

QUESTION 3