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 If then 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  