Q. 44 - Calculus

Question

In Exercises 43–46 give the first five terms for a geometric sequence ${\left\{c{r}^k\right\}}_{k=0}^{\infty }$ with the specified values of $c and r.$

$c=-2,r=-\frac{1}{3}$

Step-by-Step Solution

Answer

The five  terms are$-2,\frac{2}{3},-\frac{2}{9},\frac{2}{27},-\frac{2}{81}.$

Step 1. Given information.

The given values are $c=-2,r=-\frac{1}{3}.$

Step 2. First term.

The general sequence is $a_k=c{r}^k.$

On substituting the given values,

$a_k=(-2){\left(-\frac{1}{3}\right)}^k$

For first term,

$$\begin{gathered} k=0, \\ {a}_0=(-2){\left(-\frac{1}{3}\right)}^0=-2 \end{gathered}$$

Step 3. Remaining terms.

Now, when,

$$\begin{gathered} k=1\text{, } \\ {a}_1=(-2){\left(-\frac{1}{3}\right)}^1=\frac{2}{3} \\ k=2, \\ {a}_2=(-2){\left(-\frac{1}{3}\right)}^2=-\frac{2}{9} \\ k=3, \\ {a}_3=(-2){\left(-\frac{1}{3}\right)}^3=\frac{2}{27} \\ k=4, \\ {a}_4=(-2){\left(-\frac{1}{3}\right)}^4=-\frac{2}{81} \end{gathered}$$