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Find the common ratio.

April 7, 2026Mathematics

ScanSolve AI Answer

Answer

\text{A}

5 steps

To find the $n^{th}$ term of the series $2, 6, 18, 54, 162$ , we first determine if it is an arithmetic or geometric series.

Find the common ratio.
Divide each term by its preceding term:
$\frac{6}{2} = 3$ $\frac{18}{6} = 3$ $\frac{54}{18} = 3$ $\frac{162}{54} = 3$
Since the ratio between consecutive terms is constant, this is a geometric series with a common ratio $r = 3$ .
Identify the first term.
The first term of the series is $a = 2$ .
Use the formula for the $n^{th}$ term of a geometric series.
The formula for the $n^{th}$ term ( $T_n$ ) of a geometric series is:
$T_n = a \cdot r^{n-1}$
Substitute the values of $a$ and $r$ into the formula.
$T_n = 2 \cdot 3^{n-1}$
Compare with the given options.
The derived $n^{th}$ term matches option A.

The final answer is $\text{A}$ .