The third term of G.P. is 4. Find the product of its five terms.
Last updated date: 24th Mar 2023
•
Total views: 305.4k
•
Views today: 6.83k
Answer
305.4k+ views
Hint: Equate the third term of G.P. with 4. And convert the product of the first five terms in the form of the third term.
Complete step-by-step answer:
According to the question, the third term of the G.P. is 4. Let $a$ and $r$ be the first term and common ratio of the G.P. Then the G.P. is:
$ \Rightarrow a,ar,a{r^2},a{r^3},....$
We know that the general term of G.P. is:
${T_r} = a{r^{n - 1}}$
Third term is given as 4. So we have:
$
\Rightarrow {T_3} = 4, \\
\Rightarrow a{r^{3 - 1}} = 4, \\
\Rightarrow a{r^2} = 4 .....(i) \\
$
The product of first five terms is:
$ \Rightarrow $ Product $ = a \times ar \times a{r^2} \times a{r^3} \times a{r^4} = {a^5}{r^{10}} = {\left( {a{r^2}} \right)^5}$
Putting the value of $a{r^2}$ from equation $(i)$, we’ll get:
$ \Rightarrow $ Product $ = {\left( 4 \right)^5} = 1024$
Thus, the product of first terms of G.P. is 1024.
Note: This can be solved by another method as:
If five numbers are in G.P. then the middle number (i.e. third number) is their geometric mean.
Third term is given as 4. So, 4 is the geometric mean of the first five terms of G.P.
And if the geometric mean of five numbers is 4, then their product is ${4^5}$.
Complete step-by-step answer:
According to the question, the third term of the G.P. is 4. Let $a$ and $r$ be the first term and common ratio of the G.P. Then the G.P. is:
$ \Rightarrow a,ar,a{r^2},a{r^3},....$
We know that the general term of G.P. is:
${T_r} = a{r^{n - 1}}$
Third term is given as 4. So we have:
$
\Rightarrow {T_3} = 4, \\
\Rightarrow a{r^{3 - 1}} = 4, \\
\Rightarrow a{r^2} = 4 .....(i) \\
$
The product of first five terms is:
$ \Rightarrow $ Product $ = a \times ar \times a{r^2} \times a{r^3} \times a{r^4} = {a^5}{r^{10}} = {\left( {a{r^2}} \right)^5}$
Putting the value of $a{r^2}$ from equation $(i)$, we’ll get:
$ \Rightarrow $ Product $ = {\left( 4 \right)^5} = 1024$
Thus, the product of first terms of G.P. is 1024.
Note: This can be solved by another method as:
If five numbers are in G.P. then the middle number (i.e. third number) is their geometric mean.
Third term is given as 4. So, 4 is the geometric mean of the first five terms of G.P.
And if the geometric mean of five numbers is 4, then their product is ${4^5}$.
Recently Updated Pages
If a spring has a period T and is cut into the n equal class 11 physics CBSE

A planet moves around the sun in nearly circular orbit class 11 physics CBSE

In any triangle AB2 BC4 CA3 and D is the midpoint of class 11 maths JEE_Main

In a Delta ABC 2asin dfracAB+C2 is equal to IIT Screening class 11 maths JEE_Main

If in aDelta ABCangle A 45circ angle C 60circ then class 11 maths JEE_Main

If in a triangle rmABC side a sqrt 3 + 1rmcm and angle class 11 maths JEE_Main

Trending doubts
Difference Between Plant Cell and Animal Cell

Write an application to the principal requesting five class 10 english CBSE

Ray optics is valid when characteristic dimensions class 12 physics CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

Write the 6 fundamental rights of India and explain in detail

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

List out three methods of soil conservation

Fill in the blanks A 1 lakh ten thousand B 1 million class 9 maths CBSE

Epipetalous and syngenesious stamens occur in aSolanaceae class 11 biology CBSE
