Q. 43 - Calculus

Question

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

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

Step-by-Step Solution

Answer

The first five terms are $3,\frac{3}{2},\frac{3}{4},\frac{3}{8},\frac{3}{16}$.

Step 1. Given information.

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

Step 2. First term.

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

On substituting the given values,

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

For first term, $k=0,$

$a_0=\left(3\right){\left(\frac{1}{2}\right)}^0=3$

Step 3. Remaining terms.

Now, when,

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