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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?
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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.
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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?
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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?.
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