Question

In Problems 29–36, the given pattern continues. Write down the nth term of a sequence $\left\{a_n\right\}$suggested by the pattern.

$1,\frac{1}{2},3,\frac{1}{4},5,\frac{1}{6},7,\frac{1}{8},...$

Short answer

The nth term of the given sequence is$a_n=n$or$a_n=\frac{1}{n}$.

Step-by-step solution

Step 1. Write the given information.

The given sequence is:

$1,\frac{1}{2},3,\frac{1}{4},5,\frac{1}{6},7,\frac{1}{8},...$

Step 2. Determine the pattern of the sequence and write down the nth term.

$a_1=1,a_2=\frac{1}{2},a_3=3,a_4=\frac{1}{4},a_5=5,a_6=\frac{1}{6},a_7=7,a_8=\frac{1}{8}$

We can see from the given sequence that the terms at even place represent the reciprocal of that position for example $a_2=\frac{1}{2},a_4=\frac{1}{4}$and the terms at odd position represent the position number itself, $a_1=1,a_3=3$.

Therefore, if $n$appears to be odd then the nth term will be $n$.

And if it is at even place then the nth term will be $\frac{1}{n}$.

$a_n=n$or$a_n=\frac{1}{n}$