Find the $6^{\text {th }}$ term from the end of GP $8,4,2 \ldots \frac{1}{1024} .$
The given GP is $8,4,2 \ldots \frac{1}{1024} \cdot \rightarrow(1)$
First term in the GP, $a_{1}=a=8$
Second term in the GP, $a_{2}=a r=4$
The common ratio, $r=\frac{4}{8}=\frac{1}{2}$
The last term in the given GP is $\frac{1}{1024}$.
Second last term in the GP $=a_{n-1}=a r^{n-2}$
Starting from the end, the series forms another GP in the form,
$a r^{n-1}, a r^{n-2}, a r^{n-3} \ldots a r^{3}, a r^{2}, a r, a \rightarrow(2)$
Common ratio of this GP is $\frac{1}{r}$.
So, common ratio = 2
$a=\frac{1}{1024}$
So, $6^{\text {th }}$ term of the GP $(2)$,
$a 6=a r^{5}$
$=\frac{1}{1024} \times 2^{5}=\frac{1}{32}$
Hence, the $6^{\text {th }}$ term from the end of the given GP is $\frac{1}{32}$.
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