Chapter 9, Problem 56SGR

To determine

Write the equation for the nth term of the geometric sequence.

Answer to Problem 56SGR

The nth term of the series is: $a_n=3^n$

Explanation of Solution

Given:

The sequence: $3,9,27,\mathrm{......}$

Concept Used:

In mathematics, a geometric progression, also known as a geometric sequence, is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio.

For example, the sequence 2, 6, 18, 54, ... is a geometric progression with common ratio 3.

The ration of 2^nd term and first term is called the common ratio.

Calculation:

The sequence: $3,9,27,\mathrm{......}$

Each term in a geometric sequence can be expressed in terms of the first term $a_1$ and the common ratio $r$ . Since each succeeding term is formulated from one or more previous terms, this is a recursive formula. First term $a_1=3$ and the common difference: $r=\frac{a_2}{a_1}=\frac{9}{3}=3$

Formula for nth term of GP series is: $a_n=a_1·r^{n-1}$

Exponent rule: $\text{Product}\text{rule}=a^m·a^n=a^{m+n}$

The nth term of the series:

  $\begin{array}{}$ a_n=a_1·r^{n-1}$$ a_n=3·{(3)}^{n-1}=3^1·3^{n-1}=3^{1+n-1}=3^n$\end{array}$  $\text{Product}\text{rule}=a^m·a^n=a^{m+n}$

The nth term of the series $3,9,27,\mathrm{......}$ is $a_n=3^n$

Thus, the nth term of the series is: $a_n=3^n$