Find the 4th term from the end of the G.P. ½, 1/6, 1/18, 1/54, … , 1/4374

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Find the 4th term from the end of the G.P. ½, 1/6, 1/18, 1/54, … , 1/4374

Solution:

$n^{th}$ term from the end is given by:
$a_{n}=I(1 / r)^{n-1}$ where, $I$ is the last term, $r$ is the common ratio, $n$ is the $n^{th}$ term
Given that, last term, $I=1 / 4374$
$\begin{array}{l} r=t_{2} / t_{1}=(1 / 6) /(1 / 2) \\ =1 / 6 \times 2 / 1 \\ =1 / 3 \\ n=4 \end{array}$
Therefore, $a_{n}=\mid(1 / r)^{n-1}$
$\begin{array}{l} a_{4}=1 / 4374(1 /(1 / 3))^{4-1} \\ =1 / 4374(3 / 1)^{3} \\ =1 / 4374 \times 3^{3} \\ =1 / 4374 \times 27 \\ =1 / 162 \end{array}$
Hence, $4^{\text {th }}$ term from last is $1 / 162$ .