How Do You Find the Nth Term of a Geometric Sequence?

Use the formula $u_n=u_1r^{n-1}$, where $u_1$ is the first term, $r$ is the common ratio, and $n$ is the term number. Substitute these values and simplify to find the required term.

The Method

In a geometric sequence, each term is obtained by multiplying the preceding term by a constant called the common ratio. If two consecutive terms are known, calculate it using

$$r=\frac{u_{k+1}}{u_k}.$$

The nth term formula is

$$u_n=u_1r^{n-1}.$$

The exponent is $n-1$ because reaching the first term requires zero multiplications by $r$, reaching the second requires one multiplication, and reaching the nth term requires $n-1$ multiplications.

Worked Example

Consider the sequence

$$5,\ 15,\ 45,\ 135,\ldots$$

First, identify the initial term:

$$u_1=5.$$

Next, calculate the common ratio:

$$r=\frac{15}{5}=3.$$

Therefore, the nth term is

$$u_n=5(3)^{n-1}.$$

To find the sixth term, substitute $n=6$:

$$u_6=5(3)^5=1215.$$

A negative common ratio is also possible. For example, $2,-6,18,-54,\ldots$ has $r=-3$, so its nth term is $u_n=2(-3)^{n-1}$. The brackets are essential because the entire negative ratio is raised to the power.

Sometimes the first term is not shown directly. If $u_2=12$ and $u_5=324$, compare the terms:

$$\frac{u_5}{u_2}=r^{5-2}.$$

Moving from term 2 to term 5 involves three multiplications by $r$, so $r^3=324/12=27$ and $r=3$. Then $$, giving . Check: , which fits the ratio.

The nth-term formula is an explicit formula: it gives any term directly. By contrast, the recursive form $u_{n+1}=ru_n$ describes how to generate the next term, but reaching a distant term requires calculating all preceding terms.

Exam Technique

This is IB Mathematics: Analysis and Approaches, SL 1.3, and may appear on Paper 1 or Paper 2. Show how you found $u_1$ and $r$, then state the formula before substituting.

The common misconception is writing $u_n=u_1r^n$. Check your expression by substituting $n=1$: a correct formula must return .