Geometric Sequences

Geometric sequences are sequences where the previous number is multiplied by a certain number to obtain the next number.

The nth term of a Geometric Sequence

The nth term of a geometric sequence is shown in Figure 1.

Figure 1

Here a is the first term in the sequence, r is the common ratio and n is the term number.

Consider the sequence

                                  2, 6, 18, 54, ...

The first term is 2. The common ratio is 6 ÷ 2, or 3. In other words, each term is three times larger than the previous term.

Thus the nth term of this sequence would be 2 x 3^n-1.

The Sum of a Geometric Sequence

The formula to find the sum of a geometric sequence is shown in Figure 2.

Figure 2

The same applies here where a is the first term, r is the common ratio and n is the number of numbers being summed.

Consider the previous sequence.

If we wanted to add up the first ten numbers in the sequence, we would use this formula.

a would be 2, r would be 3 and n would be 10.

Putting these values into the formula shows us that:

The sum of the first ten numbers would then be 59048.

The Sum to Infinity

The sum to infinity of converging sequences can be represented by this formula.

Consider the sequence

                                  1, 0.5, 0.25, 0.125, ...
This sequence tends to 0. The first term is 1 and the common ratio is 0.25 ÷ 0.5, or 0.5.

Putting these values into the formula shows us that:

This shows us that the sum to infinity of this sequence is 2.

Pictures from: http://www.mathcentre.ac.uk/resources/uploaded/mc-ty-apgp-2009-1.pdf