10.5 Geometric sequences — Learn

Prerequisites: multiplication, division, and substitution into a rule.

Objectives: recognise a geometric sequence and find its term-to-term multiplier. This section develops the Higher content in A24.

A constant multiplier

A geometric sequence is formed by multiplying each term by the same number. That number is the common multiplier.

In $3,6,12,24,\ldots$, each term is multiplied by $2$. The common multiplier is $2$.

Example 1. Find the next two terms of $5,15,45,\ldots$.

The common multiplier is $3$, and the next terms are $135$ and $405$.

A multiplier below one

A multiplier can be a fraction or decimal. Multiplying by a number between zero and one makes positive terms smaller.

Example 2. Continue $80,40,20,\ldots$.

The common multiplier is $\frac12$, so the next terms are $10$ and $5$.

Progress check. Pattern B starts with $2$ tiles and doubles each time. By design 5 it has $2,4,8,16,32$ tiles.

Investigate the multiplier

Predict how changing the multiplier will affect the fifth term. Then adjust it in the figure, test your prediction, and explain the pattern you observed.

Explain: What happens to later terms when the multiplier is above $1$, equal to $1$, or between $0$ and $1$?

Watch out. Do not add the multiplier. In $3,6,12,\ldots$, the change is not a fixed amount. Check each term by dividing it by the previous term.

You should now be able to

• recognise a geometric sequence by its constant multiplier;
• find the multiplier by dividing consecutive terms;
• continue a geometric sequence with integer or fractional multipliers;
• describe how the multiplier affects later positive terms.