Question

Terms of a Sequence Find the first four terms and the 100th term of the sequence whose nth term is given.

$a_n={\left(\frac{-1}{3}\right)}^n$

Short answer

The first four terms are $-\frac{1}{3}, \frac{1}{9}, -\frac{1}{27}, \frac{1}{81}$ and the 100^th term is $\frac{1}{3^{100}}$.

Step-by-step solution

Step 1.  Given

The given sequence is $a_n={\left(\frac{-1}{3}\right)}^n$.

Step 2.  To determine

We have to find the first four terms and the 100th term of the given sequence.

Step 3.  Calculation

To find the first four terms and the 100^th term of the sequence we’ll plug n = 1, 2, 3, 4 and 100 respectively in the given sequence.

$\begin{array}{l}\text{For}n=1,a_1={\left(\frac{-1}{3}\right)}^1=-\frac{1}{3}\\ \text{For}n=2,a_2={\left(\frac{-1}{3}\right)}^2=\frac{1}{9}\\ \text{For}n=3,a_3={\left(\frac{-1}{3}\right)}^3=-\frac{1}{27}\\ \text{For}n=4,a_4={\left(\frac{-1}{3}\right)}^4=\frac{1}{81}\\ \text{For}n=100,a_{100}={\left(\frac{-1}{3}\right)}^{100}=\frac{1}{3^{100}}\end{array}$

Hence, the first four terms are $-\frac{1}{3}, \frac{1}{9}, -\frac{1}{27}, \frac{1}{81}$ and the 100^th term is $\frac{1}{3^{100}}$.