Question

What is the product of first \[2n + 1\] terms of a geometric progression?
A. The \[{\left( {n + 1} \right)^{th}}\] power of the nth term of the GP
B. The \[{\left( {2n + 1} \right)^{th}}\] power of the nth term of the GP
C. The \[{\left( {2n + 1} \right)^{th}}\] power of the \[{\left( {n{\rm{ }} + {\rm{ }}1} \right)^{th}}\] term of the GP
D. The nth power of the \[{\left( {n{\rm{ }} + {\rm{ }}1} \right)^{th}}\] terms of the GP

Answer 1

Hint: We have already known that the nth term of a Geometric progression series is \[{T_n}\; = {\rm{ }}a{r^{n - 1}}\] and here first term is ‘a’ common ratio is ‘r’. In this case, we are asked to determine the product of first \[2n + 1\] terms of a geometric progression. For that we have to write the product of geometric series and then we have to reduce the particular series using general formula to obtain the given condition

Formula Used: Geometric progression can be determined by
\[{T_n}\; = {\rm{ }}a{r^{n - 1}}\]

Complete step by step solution: We have been provided in the question that the geometric progression is
\[2n + 1\]
And we are asked to determine the product of first \[2n + 1\] terms of a geometric progression.
We have been already known that,
The Geometric progression is,
\[a,{\rm{ }}ar,{\rm{ }}a{r^2}, \ldots \ldots ..{\rm{ }}a{r^{2n}}\]
And we have been known that the product of the given geometric sequence is\[P = a \cdot ar \cdot a{r^{2\;}} \cdot a{r^3} \ldots \ldots \ldots ..{\rm{ }}a{r^{2n}}\]
Now, we have to reduce the above expression as,
\[ = {\rm{ }}{a^{2n + 1}}\;{r^{1 + 2 + \ldots + 2n}}\]
Now, we have reduce the above expression using the formula we get
\[ = {\rm{ }}{a^{2n + 1}}\;{r^{\dfrac{{2n\left( {2n + 1} \right)}}{2}}}\]
On canceling the similar terms in the powers of ‘r’ we get
\[ = {\rm{ }}{a^{2n + 1}}\;{r^{n(2n + 1)}}\]
Now, we have to multiply the bases and have the same exponent, we have
\[ = {\rm{ }}{(a{r^n})^{2n + 1}}\]
Therefore, the product of first \[2n + 1\] terms of a geometric progression is the \[{\left( {2n + 1} \right)^{th}}\] power of the \[{\left( {n{\rm{ }} + {\rm{ }}1} \right)^{th}}\] term of the GP

Option ‘C’ is correct

Note: When answering this question, be cautious not to find the sum of the geometric sequence using the formula of the first ‘n’ terms of a geometric progression. This is the most important aspect of the question; some students attempt to solve the equation immediately and wind up with a complex solution, which is time intensive and frequently results in an incorrect answer.