Chapter 12.3, Problem 22E

To determine

Calculate the first five terms of the given sequence $a_n=4+3^n$ and find whether it is geometric. If it is geometric then also calculate the common ratio, and express the $n^{th}$ term of the sequence in the standard form $a_n=a{r}^{n-1}$ .

Answer to Problem 22E

The first five term of the given sequence $a_n=4+3^n$ is $7,13,31,85\&247$

The given sequence is not in geometric

Explanation of Solution

Given information:

It is given in the question that,

the given sequence is $a_n=4+3^n$

Formula used:

  $a_n=a{r}^{n-1}$ & $r\Rightarrow \frac{a_n}{a_{n-1}}=\frac{a_{n+1}}{a_n}$

Where,

  $\begin{array}{l}aisthe{1}^{st}term\\ nisthenumberofterm\\ risthecommonratio\end{array}$

Calculation:

For finding the first five term put $n=1,2,3,4,5$ in the given sequence is $a_n=4+3^n$

So,

By putting $n=1$ in $a_n=4+3^n$

  $\begin{array}{l}{a}_1=4+3^1\\ =4+3\\ =7\end{array}$

By putting $n=2$ in $a_n=4+3^n$

  $\begin{array}{l}{a}_2=4+3^2\\ =4+9\\ =13\end{array}$

By putting $n=3$ in $a_n=4+3^n$

  $\begin{array}{l}{a}_3=4+3^3\\ =4+27\\ =31\end{array}$

By putting $n=4$ in $a_n=4+3^n$

  $\begin{array}{l}{a}_4=4+3^4\\ =4+81\\ =85\end{array}$

By putting $n=5$ in $a_n=4+3^n$

  $\begin{array}{l}{a}_5=4+3^5\\ =4+243\\ =247\end{array}$

By using the formula $r\Rightarrow \frac{a_n}{a_{n-1}}=\frac{a_{n+1}}{a_n}$

  $\begin{array}{l}r\Rightarrow \frac{a_n}{a_{n-1}}=\frac{a_{n+1}}{a_n}\\ \Rightarrow \frac{13}{7}=\frac{31}{13}\end{array}$

Here, the sequence not satisfied the given formula concept so the given sequence $a_n=4+3^n$ which shows that the common ratio of two consecutive term is not same.

So, the given sequence is not in geometric.