Q. 28 - Calculus

Question

In Exercises, find a plausible formula for the general term of the given sequence.

$\left\{5,-\frac{5}{2},\frac{5}{4},-\frac{5}{8},\frac{5}{16},\dots \right\}$

Step-by-Step Solution

Answer

The plausible formula for the general term of the sequence is $a_k=5{\left(\frac{-1}{2}\right)}^k$where $k\ge 0.$

Step 1. Given information.

The given sequence is $\left\{5,-\frac{5}{2},\frac{5}{4},-\frac{5}{8},\frac{5}{16},\dots \right\}.$

Step 2. plausible formula for the general term.

write the sequence in a pattern. 

$$\begin{gathered} \left\{5,-\frac{5}{2},\frac{5}{4},-\frac{5}{8},\frac{5}{16},\dots \right\}=\left\{\frac{5{(-1)}^0}{2^0},\frac{5{(-1)}^1}{2^1},\frac{5{(-1)}^2}{2^2},\frac{5{(-1)}^3}{2^3},\frac{5{(-1)}^4}{2^4},\dots \right\} \\ \left\{5,-\frac{5}{2},\frac{5}{4},-\frac{5}{8},\frac{5}{16},\dots \right\}={\left\{\frac{5{(-1)}^k}{2^k}\right\}}_{k=0}^{\infty } \end{gathered}$$

for every term, the numerator is $5{(-1)}^k$and the denominator is $2^k$ and $k\ge 0.$

So plausible formula for the general term of the sequence is  $a_k=\frac{5{(-1)}^k}{2^k}=5{\left(\frac{-1}{2}\right)}^k$