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If \[{5^{th}}\] term of a G.P. is 2, then the product of its 9 terms is,
A. 256
B. 512
C. 1024
D. None of these

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Last updated date: 13th Jun 2024
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Answer
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Hint: Given is only the fifth term of the G.P. and asked to find the product of 9 terms. We will consider the first term as a and the r be the common ratio of the whole G.P. then we can simply equate the fifth term as \[a{r^4} = 2\] . Then remaining 9 terms will be multiplied such that their product will be in the form of \[a \times ar \times a{r^2} \times a{r^3} \times a{r^4}.... \times a{r^8}\] and we will try to adjust this in the form of fifth term. So let’s solve it!

Complete Step by Step Solution:
Given that
\[{5^{th}}\] term of a G.P. is 2
Let first term as a and the r be the common ratio of the whole G.P.
Then fifth term will be \[a{r^4} = 2\]
But they are asked to find the product of 9 terms of the G.P.
So we can write the product as \[a \times ar \times a{r^2} \times a{r^3} \times a{r^4}.... \times a{r^8}\]
So ,
\[ \Rightarrow a \times ar \times a{r^2} \times a{r^3} \times a{r^4} \times a{r^5} \times a{r^6} \times a{r^7} \times a{r^8}\]
Now we can write the terms with base a separately and those with r separately.
\[ \Rightarrow aaaaaaaaa \times r{r^2}{r^3}{r^4}{r^5}{r^6}{r^7}{r^8}\]
Now adding the powers of the bases separately,
\[ \Rightarrow {a^{1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1}} \times {r^{1 + 2 + 3 + 4 + 5 + 6 + 7 + 8}}\]
Now we can write as,
\[ \Rightarrow {a^9} \times {r^{36}}\]
But we have to write this in the form of power 4,
\[ \Rightarrow {a^9} \times {r^{9 \times 4}}\]
Now we will take the common power out,
\[ \Rightarrow {\left( {a{r^4}} \right)^9}\]
Putting the value of fifth term in above bracket,
\[ \Rightarrow {\left( 2 \right)^9}\]
Then the value of the ninth power of 2 is our answer.
\[ \Rightarrow 521\]
This is our final answer.

Note:
Here note that the fifth term is having fourth power of 2 and not fifth power. We need not to find all nine terms separately; only finding the product is enough because that product will then be written in the form of the term that is known. Terms in a G.P. are having a common ratio in between. That’s why the power of r is increasing as the terms are increasing.