13.3 Geometric sequences  (Page 3/6)

Writing an explicit formula for the n Th term of a geometric sequence

Write an explicit formula for the $n\text{th}$ term of the following geometric sequence.

$$\left\{2\text{, }10\text{, }50\text{, }250\text{, }\mathrm{...}\right\}$$

The first term is 2. The common ratio can be found by dividing the second term by the first term.

$$\frac{10}{2}=5$$

The common ratio is 5. Substitute the common ratio and the first term of the sequence into the formula.

$$\begin{array}{l}{a}_n=a_1{r}^{(n-1)}\hfill \\ a_n=2\cdot 5^{n-1}\hfill \end{array}$$

The graph of this sequence in [link] shows an exponential pattern.

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Solving application problems with geometric sequences

In real-world scenarios involving arithmetic sequences, we may need to use an initial term of $a_0$ instead of $a_1.$ In these problems, we can alter the explicit formula slightly by using the following formula:

Key equations

Key concepts

• A geometric sequence is a sequence in which the ratio between any two consecutive terms is a constant.
• The constant ratio between two consecutive terms is called the common ratio.
• The common ratio can be found by dividing any term in the sequence by the previous term. See [link] .
• The terms of a geometric sequence can be found by beginning with the first term and multiplying by the common ratio repeatedly. See [link] and [link] .
• A recursive formula for a geometric sequence with common ratio $r$ is given by $a_n=r{a}_{n–1}$ for $n\ge 2$ .
• As with any recursive formula, the initial term of the sequence must be given. See [link] .
• An explicit formula for a geometric sequence with common ratio $r$ is given by $a_n=a_1{r}^{n–1}.$ See [link] .
• In application problems, we sometimes alter the explicit formula slightly to $a_n=a_0{r}^n.$ See [link] .