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Which term of the $G.P. \space 4, 64, 1024, ....$ upto $n$ terms is $262144 ?$

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Answer:

$5 ^ { th }$

Step by Step Explanation:

1. A geometric progression $(G.P.)$ is of the form, $a, ar, ar^2, ar^3, ......,$ where $a$ is called the first term and $r$ is called the common ratio of the $G.P.$
   The $n^{ th }$ term of a $G.P.$ is given by, $a_n= ar^{n-1}$
2. Let $262144$ be the $n^{ th }$ term of the given $G.P.,$ so, we need to find the value of $n.$
   Here, the first term, $a = 4$
   The common ratio, $r = \dfrac{ a_{k+1} }{ a_k }$ where $k \ge 1$
   $\implies r = \dfrac{a_{1+1} }{ a_1 } = \dfrac{ a_2 }{ a_1 } = \dfrac{ 64 }{ 4 } = 16$
3. Now, $$\begin{align} & a_{ n } = 262144 \\ \implies & ar^{ n - 1 } = 262144 \\ \implies & 4(16)^{ n-1 } = 262144 \\ \implies & 16^{ n-1 } = \dfrac{ 262144 } { 4 } \\ \implies & 16^{ n-1 } = 65536 \\ \implies & 16^{ n-1 } = 16^{ 4 } \\ \implies & n - 1 = 4 \\ \implies & n = 4 + 1 \\ \implies & n = 5 \\ \end{align}$$
4. Hence, the $5^{th}$ term of the given $G.P.$ is $262144$.

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