What is the geometric sequence formula?

There are two types of sequences in the IB mathematics course syllabi; arithmetic sequences and geometric sequences. A geometric sequence is one where the next term is obtained by multiplying the previous term by a fixed number called the common ratio.

There are two formulas associated with geometric sequences. The first formula, shown below, is used to find a term value in the sequence.

$$\begin{align*} u_n&=u_1\times r^{n-1} \end{align*}$$

This means if we know the first value in a geometric sequence $(u_1)$ and the common ratio $(r)$ we can find any value in the sequence based on their position in the sequence (i.e. $3$rd term, $10$th term, etc.).

For example, if we want to find the $5$th term in a geometric sequence with the first term of $2$ and a common ratio of $3$ we use the given formula with $n=5$, $u_1=2$ and $r=3$.

$$\begin{align*} u_5&=2(3)^{5-1}\\[6pt] &=2(3)^4\\[6pt] &=162 \end{align*}$$

This means that the $5$th term in the sequence is $162$.

The other formula associated with geometric sequences is the sum formula.

$$\begin{align*} S_n&=\frac{u_1(r^n-1)}{r-1}=\frac{u_1(1-r^n)}{1-r} \end{align*}$$

We use this formula when we want to $\textbf{add}$ all the numbers in a geometric sequence. This sum of the geometric sequence is also called a geometric series.

Notice that this formula is written in two ways. Both forms of the equation will get you to the correct answer but generally it is easier to use the first form when $|r|>1$ and the second form when $|r|<1$.

Let’s look at the sequence above where $n=5$, $u_1=2$ and $r=3$. If we use the sum formula (choosing the first form, as we have $r>1$)

$$\begin{align*} S_5&=\frac{2(3^5-1)}{3-1}\\[6pt] &=\frac{\cancel{2}(243-1)}{\cancel{2}}\\[6pt] &=242 \end{align*}$$

We find that the sum of the first $5$ terms is $162$.

This is the same result we would get if we generated the first $5$ terms in the sequence and then added them together.

$$\begin{align*} S_5&=2+6+18+54+162\\[6pt] &=242 \end{align*}$$

There is also a formula for the sum to infinity of a geometric sequence which is

$$\begin{align*} S_{\infty}&=\frac{u_1}{1-r}\hspace{1em}\text{where }|r|<1 \end{align*}$$

This formula appears in the Mathematics AA HL, AA SL and AI HL courses. It is not in the AI SL course.

We use this formula when $|r|<1$ because in this case, each successive term added to the sum is smaller (so the sum will approach some value that we can find). Alternatively, when $|r|>1$ then we would be adding larger numbers each time and the sum to infinity would also be infinite.