Answer

Verified

373.2k+ views

**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.

Recently Updated Pages

In the equation x2+y2+2gx+2fy+c0 represents a circle class 10 maths CBSE

How do you Factor completely 2x3 + 5x2 37x 60 class 10 maths CBSE

How do you prove the identity dfrac11 cos x + dfrac11 class 10 maths CBSE

How do you solve 2x2 5x 12 0 by completing the squ class 10 maths CBSE

How do you solve 2x+46 class 10 maths CBSE

How do you simplify dfracm42m4 and write it using only class 10 maths CBSE

Trending doubts

Difference Between Plant Cell and Animal Cell

Give 10 examples for herbs , shrubs , climbers , creepers

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Write a letter to the principal requesting him to grant class 10 english CBSE

Change the following sentences into negative and interrogative class 10 english CBSE

Write the 6 fundamental rights of India and explain in detail

Name 10 Living and Non living things class 9 biology CBSE

The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths