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A G.P(Geometric Progression)consists of 2n terms. If the sum of the terms occupying the odd places is S1 and that of the terms in even places is S2 , the common ratio of the progression is
- a)n
- b)2S1
- c)S2/S1
- d)S1/S2
Correct answer is option 'C'. Can you explain this answer?
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Most Upvoted Answer
A G.P(Geometric Progression)consists of 2n terms. If the sum of the te...
Solution:
Let the first term of the G.P be 'a' and the common ratio be 'r'.
Then, the terms in odd places are: a, ar^2, ar^4, ..., ar^(2n-2)
And, the terms in even places are: ar, ar^3, ar^5, ..., ar^(2n-1)
Sum of terms in odd places (S1):
S1 = a + ar^2 + ar^4 + ... + ar^(2n-2)
S1 = a(1 + r^2 + r^4 + ... + r^(2n-2)) (sum of n terms of G.P with common ratio r^2)
S1 = a(1 - r^(2n))/(1 - r^2)
Sum of terms in even places (S2):
S2 = ar + ar^3 + ar^5 + ... + ar^(2n-1)
S2 = ar(1 + r^2 + r^4 + ... + r^(2n-2)) (sum of n terms of G.P with common ratio r^2)
S2 = ar(1 - r^(2n))/(1 - r^2)
We need to find the common ratio 'r' of the G.P.
Using the formula for S1 and S2, we can write:
S1/S2 = (a(1 - r^(2n))/(1 - r^2))/(ar(1 - r^(2n))/(1 - r^2))
S1/S2 = a/r
Multiplying both sides by r, we get:
S1/S2 * r = a
Substituting this value of 'a' in the equation for S1, we get:
S1 = S1/S2 * r * (1 - r^(2n))/(1 - r^2)
Dividing this equation by the equation for S2, we get:
S1/S2 = (S1/S2 * r * (1 - r^(2n))/(1 - r^2))/ar(1 - r^(2n))/(1 - r^2)
S1/S2 = r
Hence, the common ratio of the G.P is S2/S1. (option 'C')
Let the first term of the G.P be 'a' and the common ratio be 'r'.
Then, the terms in odd places are: a, ar^2, ar^4, ..., ar^(2n-2)
And, the terms in even places are: ar, ar^3, ar^5, ..., ar^(2n-1)
Sum of terms in odd places (S1):
S1 = a + ar^2 + ar^4 + ... + ar^(2n-2)
S1 = a(1 + r^2 + r^4 + ... + r^(2n-2)) (sum of n terms of G.P with common ratio r^2)
S1 = a(1 - r^(2n))/(1 - r^2)
Sum of terms in even places (S2):
S2 = ar + ar^3 + ar^5 + ... + ar^(2n-1)
S2 = ar(1 + r^2 + r^4 + ... + r^(2n-2)) (sum of n terms of G.P with common ratio r^2)
S2 = ar(1 - r^(2n))/(1 - r^2)
We need to find the common ratio 'r' of the G.P.
Using the formula for S1 and S2, we can write:
S1/S2 = (a(1 - r^(2n))/(1 - r^2))/(ar(1 - r^(2n))/(1 - r^2))
S1/S2 = a/r
Multiplying both sides by r, we get:
S1/S2 * r = a
Substituting this value of 'a' in the equation for S1, we get:
S1 = S1/S2 * r * (1 - r^(2n))/(1 - r^2)
Dividing this equation by the equation for S2, we get:
S1/S2 = (S1/S2 * r * (1 - r^(2n))/(1 - r^2))/ar(1 - r^(2n))/(1 - r^2)
S1/S2 = r
Hence, the common ratio of the G.P is S2/S1. (option 'C')
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Community Answer
A G.P(Geometric Progression)consists of 2n terms. If the sum of the te...
S1= (a+ar^2+ar^4+upto ar^2n-2)
S2= (ar+ar+ar^3 upto ar^2n-1)
Now, S2/S1= ar(1+r^2+r^4 upto r^2n-2)/a(1+1+r^2+r^4 upto r^2n-2)
therefore, S2/S1= r
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A G.P(Geometric Progression)consists of 2n terms. If the sum of the terms occupying the odd places is S1and that of the terms in even places is S2, the common ratio of the progression isa)nb)2S1c)S2/S1d)S1/S2Correct answer is option 'C'. Can you explain this answer?
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A G.P(Geometric Progression)consists of 2n terms. If the sum of the terms occupying the odd places is S1and that of the terms in even places is S2, the common ratio of the progression isa)nb)2S1c)S2/S1d)S1/S2Correct answer is option 'C'. Can you explain this answer? 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 A G.P(Geometric Progression)consists of 2n terms. If the sum of the terms occupying the odd places is S1and that of the terms in even places is S2, the common ratio of the progression isa)nb)2S1c)S2/S1d)S1/S2Correct answer is option 'C'. Can you explain this answer? covers all topics & solutions for CA Foundation 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A G.P(Geometric Progression)consists of 2n terms. If the sum of the terms occupying the odd places is S1and that of the terms in even places is S2, the common ratio of the progression isa)nb)2S1c)S2/S1d)S1/S2Correct answer is option 'C'. Can you explain this answer?.
A G.P(Geometric Progression)consists of 2n terms. If the sum of the terms occupying the odd places is S1and that of the terms in even places is S2, the common ratio of the progression isa)nb)2S1c)S2/S1d)S1/S2Correct answer is option 'C'. Can you explain this answer? 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 A G.P(Geometric Progression)consists of 2n terms. If the sum of the terms occupying the odd places is S1and that of the terms in even places is S2, the common ratio of the progression isa)nb)2S1c)S2/S1d)S1/S2Correct answer is option 'C'. Can you explain this answer? covers all topics & solutions for CA Foundation 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A G.P(Geometric Progression)consists of 2n terms. If the sum of the terms occupying the odd places is S1and that of the terms in even places is S2, the common ratio of the progression isa)nb)2S1c)S2/S1d)S1/S2Correct answer is option 'C'. Can you explain this answer?.
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