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#mathematics
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#geometric-progression
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A) The sum of infinity of GP is four times the first term. Find the common ratio?
B) The limiting sum of a GP is twice the sum of the first two terms. Find the possible values of the common ratio?
Awa Kals
Mar 24
A) The sum of infinity of GP is four times the first term. Find the common ratio? B) The limiting sum of a GP is twice the sum of the first two terms. Find the possible values of the common ratio?
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Mar 24
A) Let the first term of the GP be a and the common ratio be r.
The sum of an infinite GP is given by S = a / (1 - r)
Given that the sum is four times the first term:
4a = a / (1 - r)
4 = 1 / (1 - r)
4 - 4r = 1
4r = 3
r = 3/4
Therefore, the common ratio is 3/4.
B) Let the first two terms of the GP be a and ar.
The sum of the first two terms is given by S2 = a + ar = a(1 + r)
Given that the limiting sum is twice the sum of the first two terms:
2(a / (1 - r)) = a(1 + r)
2 / (1 - r) = 1 + r
2 = (1 - r)(1 + r)
2 = 1 - r^2
r^2 = -1
Since the common ratio of a geometric progression must be positive, there are no possible values of the common ratio that satisfy this condition.
The sum of an infinite GP is given by S = a / (1 - r)
Given that the sum is four times the first term:
4a = a / (1 - r)
4 = 1 / (1 - r)
4 - 4r = 1
4r = 3
r = 3/4
Therefore, the common ratio is 3/4.
B) Let the first two terms of the GP be a and ar.
The sum of the first two terms is given by S2 = a + ar = a(1 + r)
Given that the limiting sum is twice the sum of the first two terms:
2(a / (1 - r)) = a(1 + r)
2 / (1 - r) = 1 + r
2 = (1 - r)(1 + r)
2 = 1 - r^2
r^2 = -1
Since the common ratio of a geometric progression must be positive, there are no possible values of the common ratio that satisfy this condition.
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