
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
163.5k+ views
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.
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.
Recently Updated Pages
Geometry of Complex Numbers – Topics, Reception, Audience and Related Readings

JEE Main 2021 July 25 Shift 1 Question Paper with Answer Key

JEE Main 2021 July 22 Shift 2 Question Paper with Answer Key

JEE Main 2025 Session 2: Exam Date, Admit Card, Syllabus, & More

JEE Atomic Structure and Chemical Bonding important Concepts and Tips

JEE Amino Acids and Peptides Important Concepts and Tips for Exam Preparation

Trending doubts
Degree of Dissociation and Its Formula With Solved Example for JEE

Instantaneous Velocity - Formula based Examples for JEE

JEE Main Chemistry Question Paper with Answer Keys and Solutions

JEE Main Reservation Criteria 2025: SC, ST, EWS, and PwD Candidates

What is Normality in Chemistry?

Chemistry Electronic Configuration of D Block Elements: JEE Main 2025

Other Pages
NCERT Solutions for Class 11 Maths Chapter 6 Permutations and Combinations

NCERT Solutions for Class 11 Maths Chapter 8 Sequences and Series

Total MBBS Seats in India 2025: Government College Seat Matrix

NEET Total Marks 2025: Important Information and Key Updates

Neet Cut Off 2025 for MBBS in Tamilnadu: AIQ & State Quota Analysis

Karnataka NEET Cut off 2025 - Category Wise Cut Off Marks
