Class 11 Exam > Class 11 Questions > In a G.P. of even number of terms, the sum of...
Start Learning for Free
In a G.P. of even number of terms, the sum of all terms is 5 times the sum of odd terms. The common ratio of the G.P is?
Most Upvoted Answer
In a G.P. of even number of terms, the sum of all terms is 5 times the...
Community Answer
In a G.P. of even number of terms, the sum of all terms is 5 times the...
Sum of all terms in the G.P
- Let's assume the G.P has n terms, so the sum of all terms can be calculated using the formula:
S = a(r^n - 1) / (r - 1)
where S is the sum, a is the first term, r is the common ratio, and n is the number of terms.
Sum of odd terms in the G.P
- In a G.P with n terms, the odd terms are a, a*r^2, a*r^4, ... and the sum of these terms can be calculated using the formula:
S_odd = a(1 + r^2 + r^4 + ...) = a / (1 - r^2)
Given condition
- The sum of all terms is 5 times the sum of odd terms, so we have:
S = 5S_odd
a(r^n - 1) / (r - 1) = 5a / (1 - r^2)
Solving for common ratio (r)
- By simplifying the above equation, we get:
r^n - 1 = 5(1 - r^2)
r^n - 1 = 5 - 5r^2
r^n + 5r^2 = 6
- Since the number of terms in a G.P is even, let's assume n = 2m. Substituting this in the equation:
r^(2m) + 5r^2 = 6
r^2(1 + 5) = 6
r^2 = 6 / 6
r^2 = 1
r = 1 or -1
Conclusion
- The common ratio for the G.P can be either 1 or -1.
- Let's assume the G.P has n terms, so the sum of all terms can be calculated using the formula:
S = a(r^n - 1) / (r - 1)
where S is the sum, a is the first term, r is the common ratio, and n is the number of terms.
Sum of odd terms in the G.P
- In a G.P with n terms, the odd terms are a, a*r^2, a*r^4, ... and the sum of these terms can be calculated using the formula:
S_odd = a(1 + r^2 + r^4 + ...) = a / (1 - r^2)
Given condition
- The sum of all terms is 5 times the sum of odd terms, so we have:
S = 5S_odd
a(r^n - 1) / (r - 1) = 5a / (1 - r^2)
Solving for common ratio (r)
- By simplifying the above equation, we get:
r^n - 1 = 5(1 - r^2)
r^n - 1 = 5 - 5r^2
r^n + 5r^2 = 6
- Since the number of terms in a G.P is even, let's assume n = 2m. Substituting this in the equation:
r^(2m) + 5r^2 = 6
r^2(1 + 5) = 6
r^2 = 6 / 6
r^2 = 1
r = 1 or -1
Conclusion
- The common ratio for the G.P can be either 1 or -1.
Attention Class 11 Students!
To make sure you are not studying endlessly, EduRev has designed Class 11 study material, with Structured Courses, Videos, & Test Series. Plus get personalized analysis, doubt solving and improvement plans to achieve a great score in Class 11.
|
Explore Courses for Class 11 exam
|
|
Similar Class 11 Doubts
Top Courses for Class 11View all
In a G.P. of even number of terms, the sum of all terms is 5 times the sum of odd terms. The common ratio of the G.P is?
Question Description
In a G.P. of even number of terms, the sum of all terms is 5 times the sum of odd terms. The common ratio of the G.P is? for Class 11 2024 is part of Class 11 preparation. The Question and answers have been prepared according to the Class 11 exam syllabus. Information about In a G.P. of even number of terms, the sum of all terms is 5 times the sum of odd terms. The common ratio of the G.P is? covers all topics & solutions for Class 11 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for In a G.P. of even number of terms, the sum of all terms is 5 times the sum of odd terms. The common ratio of the G.P is?.
In a G.P. of even number of terms, the sum of all terms is 5 times the sum of odd terms. The common ratio of the G.P is? for Class 11 2024 is part of Class 11 preparation. The Question and answers have been prepared according to the Class 11 exam syllabus. Information about In a G.P. of even number of terms, the sum of all terms is 5 times the sum of odd terms. The common ratio of the G.P is? covers all topics & solutions for Class 11 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for In a G.P. of even number of terms, the sum of all terms is 5 times the sum of odd terms. The common ratio of the G.P is?.
Solutions for In a G.P. of even number of terms, the sum of all terms is 5 times the sum of odd terms. The common ratio of the G.P is? in English & in Hindi are available as part of our courses for Class 11.
Download more important topics, notes, lectures and mock test series for Class 11 Exam by signing up for free.
Here you can find the meaning of In a G.P. of even number of terms, the sum of all terms is 5 times the sum of odd terms. The common ratio of the G.P is? defined & explained in the simplest way possible. Besides giving the explanation of
In a G.P. of even number of terms, the sum of all terms is 5 times the sum of odd terms. The common ratio of the G.P is?, a detailed solution for In a G.P. of even number of terms, the sum of all terms is 5 times the sum of odd terms. The common ratio of the G.P is? has been provided alongside types of In a G.P. of even number of terms, the sum of all terms is 5 times the sum of odd terms. The common ratio of the G.P is? theory, EduRev gives you an
ample number of questions to practice In a G.P. of even number of terms, the sum of all terms is 5 times the sum of odd terms. The common ratio of the G.P is? tests, examples and also practice Class 11 tests.
|
|
Explore Courses for Class 11 exam
|
|
Signup for Free!
Signup to see your scores go up within 7 days! Learn & Practice with 1000+ FREE Notes, Videos & Tests.
|
© EduRev
|
Education Revolution
|
Follow Us
|
Signup to see your scores
go up within 7 days!
Access 1000+ FREE Docs, Videos and Tests
Takes less than 10 seconds to signup