Verify whether the following sequences are G.P. If so, write tn. … - Vidyadip

Question

Verify whether the following sequences are G.P. If so, write tn. : $2,6,18,54, \ldots \ldots$

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Answer

$2,6,18,54, \ldots \ldots$.
$\mathrm{t}_1=2, \mathrm{t}_2=6, \mathrm{t}_3=18, \mathrm{t}_4=54, \ldots$.
Here, $\frac{t_2}{t_1}=\frac{t_3}{t_2}=\frac{t_4}{t_3}=3$
Since, the ratio of any two consecutive terms is a constant, the given sequence is a geometric progression.
Here, $a=2, r=3$
$
\begin{aligned}
& \mathrm{t}_{\mathrm{n}}=\mathrm{ar} \mathrm{r}^{\mathrm{n}-1} \\
& \therefore \mathrm{t}_{\mathrm{n}}=2\left(3^{\mathrm{n}-1}\right)
\end{aligned}
$

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