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The sum of the first two terms of an infinite geometric series is 15 and each term is equal to the sum of all the terms following it.then the sum of the series is?
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The sum of the first two terms of an infinite geometric series is 15 a...
Given:
To find:
Solution:
Let the first term of the series be 'a' and the common ratio be 'r'.
Then, the given information can be expressed as:
Solving the first equation for 'a', we get:
Substituting this value of 'a' in the second equation, we get:
Multiplying both sides by (1+r) and simplifying, we get:
So, r can be either 1 or 15.
If r = 1, then a = 7.5 (from the first equation) and the sum of the series is infinite.
If r = 15, then a = 0.5 (from the first equation) and the sum of the series is:
But since the sum of a geometric series must be a positive number, we can conclude that the series diverges when r = 1 and the sum of the series when r = 15 is -1/28.
Therefore, the sum of the series is -1/28.
- The sum of the first two terms of an infinite geometric series is 15.
- Each term is equal to the sum of all the terms following it.
To find:
- The sum of the series.
Solution:
Let the first term of the series be 'a' and the common ratio be 'r'.
Then, the given information can be expressed as:
- a + ar = 15 (sum of first two terms is 15)
- a = ar + ar^2 + ar^3 + ... (each term equals the sum of all the terms following it)
Solving the first equation for 'a', we get:
- a(1+r) = 15
- a = 15/(1+r)
Substituting this value of 'a' in the second equation, we get:
- 15/(1+r) = r(1 + 15/(1+r) + 15/(1+r)^2 + ...)
Multiplying both sides by (1+r) and simplifying, we get:
- 15 = r(1 + 15/(1-r))
- 15 = r((1-r + 15)/(1-r))
- 15(1-r) = r(16-r)
- 15 - 15r = 16r - r^2
- r^2 - 16r + 15 = 0
- (r-1)(r-15) = 0
So, r can be either 1 or 15.
If r = 1, then a = 7.5 (from the first equation) and the sum of the series is infinite.
If r = 15, then a = 0.5 (from the first equation) and the sum of the series is:
- S = a/(1-r) = 0.5/(1-15) = -1/28
But since the sum of a geometric series must be a positive number, we can conclude that the series diverges when r = 1 and the sum of the series when r = 15 is -1/28.
Therefore, the sum of the series is -1/28.
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The sum of the first two terms of an infinite geometric series is 15 and each term is equal to the sum of all the terms following it.then the sum of the series is?
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The sum of the first two terms of an infinite geometric series is 15 and each term is equal to the sum of all the terms following it.then the sum of the series is? for CA Foundation 2024 is part of CA Foundation preparation. The Question and answers have been prepared according to the CA Foundation exam syllabus. Information about The sum of the first two terms of an infinite geometric series is 15 and each term is equal to the sum of all the terms following it.then the sum of the series is? covers all topics & solutions for CA Foundation 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The sum of the first two terms of an infinite geometric series is 15 and each term is equal to the sum of all the terms following it.then the sum of the series is?.
The sum of the first two terms of an infinite geometric series is 15 and each term is equal to the sum of all the terms following it.then the sum of the series is? for CA Foundation 2024 is part of CA Foundation preparation. The Question and answers have been prepared according to the CA Foundation exam syllabus. Information about The sum of the first two terms of an infinite geometric series is 15 and each term is equal to the sum of all the terms following it.then the sum of the series is? covers all topics & solutions for CA Foundation 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The sum of the first two terms of an infinite geometric series is 15 and each term is equal to the sum of all the terms following it.then the sum of the series is?.
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