Question

nth term of a Sequence Find the nth term of a sequence whose first several terms are given.

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

Short answer

The nth term of the sequence is$a_n=\left\{\begin{array}{l}n,n=odd\\ \frac{1}{n},n=even\end{array}\right..$

Step-by-step solution

Step 1.  Given

The given sequence is $1, \frac{1}{2}, 3, \frac{1}{4}, 5, \frac{1}{6},$…

Step 2.  To determine

We have to find thenth term of the sequence.

Step 3.  Calculation

The given sequence $1, \frac{1}{2}, 3, \frac{1}{4}, 5, \frac{1}{6},$…

This is a dual sequence.

It has odd positions to be 1, 3, 5, …

Here we see that:

1^st position = 1

3^rd position = 3

5^th position = 5

It indicates that for n = odd, nth term is n.

The even positions are $\frac{1}{2}, \frac{1}{4}, \frac{1}{6},...$

2^nd position = $\frac{1}{2}$.

4^th position = $\frac{1}{4}$.

6^th position = $\frac{1}{6}$

It indicates that for n = even, nth term is $\frac{1}{n}$.

So, we can write the sequence as:

$a_n=\left\{\begin{array}{l}n,n=odd\\ \frac{1}{n},n=even\end{array}\right.$

Hence, the nth term of the sequence is$a_n=\left\{\begin{array}{l}n,n=odd\\ \frac{1}{n},n=even\end{array}\right..$