Question

$2,6,18,54,162........................$ what is the nth term of the GP?
A. ${\left( 3 \right)^{n - 1}}$
B. $2{\left( 3 \right)^{n - 1}}$
C. $2{\left( 3 \right)^{n + 1}}$
D. $2{\left( 3 \right)^n}$

Answer 1

- Hint: First of all, find the first term and common ratio of the given series. Then the nth term of the series in a GP is given by ${a_n} = a{r^{n - 1}}$ where $a$ is the first term and $r$ is the common ratio of the series of $n$ terms. So, use this concept to reach the solution of the given problem.

Complete step-by-step solution -

The given series $2,6,18,54,162........................$ is in GP.
We know that the nth term of the series in a GP is given by ${a_n} = a{r^{n - 1}}$ where $a$ is the first term and $r$ is the common ratio of the series of $n$ terms.
In the given series $a = 2$
The common ratio of a series in GP is given by $\dfrac{{{a_2}}}{{{a_1}}}$.
So, the common ratio of the given series is $\dfrac{6}{2} = 3$
Therefore, the nth term of the given series is ${a_n} = 2{\left( 3 \right)^{n - 1}}$
Hence, the nth terms of the series $2,6,18,54,162........................$ is $2{\left( 3 \right)^{n - 1}}$
Thus, the correct option is B. $2{\left( 3 \right)^{n - 1}}$

Note: The common ratio of a series in GP is given by $\dfrac{{{a_2}}}{{{a_1}}}$. A geometric progression, also known as 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 common ratio.