Sequences Defined by an Explicit Formula

How To

Given an explicit formula for a piecewise function, write the first $n$ terms of a sequence

1. Identify the formula to which $n=1$ applies.

2. To find the first term, $a_1$, use $n=1$ in the appropriate formula.

3. Identify the formula to which $n=2$ applies.

4. To find the second term, $a_2$, use $n=2$ in the appropriate formula.

5. Continue in the same manner until you have identified all $n$ terms.

Example 3

Writing the Terms of a Sequence Defined by a Piecewise Explicit Formula

Write the first six terms of the sequence.

$a_{n}=\left\{\begin{array}{lll}n^{2} & \text { if } n & \text { is not divisible by } 3 \\\frac{n}{3} & \text { if } n & \text { is divisible by } 3\end{array}\right.$

Solution

Substitute $n=1,n=2$, and so on in the appropriate formula. Use $n^2$ when $n$ is not a multiple of 3. Use $\frac{n}{3}$ when $n$ is a multiple of 3.

$\begin{array}{ll}a_{1}=1^{2}=1 & 1 \text { is not a multiple of } 3 . \text { Use } n^{2} . \\a_{2}=2^{2}=4 & 2 \text { is not a multiple of } 3 . \text { Use } n^{2} . \\a_{3}=\frac{3}{3}=1 & 3 \text { is a multiple of } 3 . \text { Use } \frac{n}{3} . \\a_{4}=4^{2}=16 & 4 \text { is not a multiple of } 3 . \text { Use } n^{2} . \\a_{5}=5^{2}=25 & 5 \text { is not a multiple of } 3 . \text { Use } n^{2} . \\a_{6}=\frac{6}{3}=2 & 6 \text { is a. multiple of } 3 . \text { Use } \frac{n}{3} .\end{array}$

The first six terms are $\{1,4,1,16,25,2\}$.

Analysis

Every third point on the graph shown in Figure 4 stands out from the two nearby points. This occurs because the sequence was defined by a piecewise function.

Figure 4

Try It #3

Write the first six terms of the sequence.

$a_{n}=\left\{\begin{array}{lll}2n^{3} & \text { if } n & \text { is odd } \\\frac{5n}{2} & \text { if } n & \text { is even }\end{array}\right.$