Some problems involving geometric sequences involve much manipulation.
Example: The first three terms of a geometric sequence are
and
Find
the first term and the common ratio.
The common ratio is equal to the second term divided by the first and also equal to the third term divided by the second, hence
Cross multiplication gives
Expanding both sides and simplifying gives
Hence
so
or
If
the first three terms are 1,-2, 4. The first term is 1 and the common ratio is -2
If
the first three terms are 5, 10, 20. The first term is 5 and the common ratio is 2.
Example: The first first, second and fourth terms of a geometric sequence are the first, second and third terms of an arithmetic sequence. Find the common ratio of the geometric sequence.
The first four terms of the geometric sequence may be written
so the first second and fourth terms are
Since these are the first, second and third terms of an arithmetic sequence,
Moving
to the right hand side,becomes a common factor so we can factorise with
Hence
or
is a root of
Ifthen the terms of the sequence are all the same. The common difference is then 0.
If
then the terms of the sequence will alternate in sign so there cannot be a single number added to each term to give the next term, so