Q. 46 - 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 $c and r.$

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

Step-by-Step Solution

Answer

The first five terms are $-1,\frac{1}{2},-\frac{1}{4},\frac{1}{8},-\frac{1}{16}$.

Step 1. Given information.

The given data is $$\begin{gathered} c=-1, \\ r=-\frac{1}{2}. \end{gathered}$$

Step 2. First term.

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

Substituting the given values,

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

For the first term $k=0,$

Therefore,

$$\begin{gathered} a_0=(-1){\left(-\frac{1}{2}\right)}^0 \\ =-1 \end{gathered}$$

Step 3. Remaining terms.

Substituting,

$$\begin{gathered} k=1, \\ {a}_1=(-1){\left(-\frac{1}{2}\right)}^1=\frac{1}{2} \\ k=2 \\ {a}_2=(-1){\left(-\frac{1}{2}\right)}^2=-\frac{1}{4} \\ k=3, \\ {a}_3=(-1){\left(-\frac{1}{2}\right)}^3=\frac{1}{8} \\ k=4 \\ {a}_4=(-1){\left(-\frac{1}{2}\right)}^4=-\frac{1}{16} \end{gathered}$$