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.