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If the first term of a G.P exceeds the second term by 2 and the sum to infinity is 50 the series is ________.
- a)10, 8, 32/5 …
- b)10, 8, 5/2…
- c)10, 10/3, 10/9…
- d)None
Correct answer is option 'A'. Can you explain this answer?
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If the first term of a G.P exceeds the second term by 2 and the sum to...
Let the second term be x, then the first term is x+2.
The sum to infinity of a G.P with first term a and common ratio r is given by:
S = a/(1-r)
Here, S = 50 and a = x+2. Since the G.P is unspecified, we don't know the value of r.
However, we can use the fact that the second term is x to write:
x = a*r
Substituting for a, we have:
x = (x+2)*r
Solving for r, we get:
r = x/(x+2)
Substituting this into the formula for S, we have:
50 = (x+2)/(1-x/(x+2))
Simplifying this expression, we get:
50 = (x+2)^2/(2x+2)
Multiplying both sides by 2x+2, we get:
50(2x+2) = (x+2)^2
Expanding the right side, we get:
50(2x+2) = x^2 + 4x + 4
Simplifying and rearranging, we get:
x^2 - 92x + 96 = 0
Solving for x using the quadratic formula, we get:
x = (92 ± √(92^2 - 4*96))/2
x ≈ 1.043 or x ≈ 90.957
Since the common ratio of a G.P must be between -1 and 1 for the sum to converge, we must have x ≈ 1.043.
Therefore, the first term is x+2 ≈ 3.043 and the common ratio is r ≈ 1.043/3.043 ≈ 0.342.
The G.P is 3.043, 1.043, 0.356, 0.122, 0.042, ...
The sum of the first 5 terms is 3.043+1.043+0.356+0.122+0.042 = 4.606.
Taking the limit as n goes to infinity, we see that the sum approaches 4.606/(1-0.342) = 6.975.
Therefore, the answer is (E) none of the above.
The sum to infinity of a G.P with first term a and common ratio r is given by:
S = a/(1-r)
Here, S = 50 and a = x+2. Since the G.P is unspecified, we don't know the value of r.
However, we can use the fact that the second term is x to write:
x = a*r
Substituting for a, we have:
x = (x+2)*r
Solving for r, we get:
r = x/(x+2)
Substituting this into the formula for S, we have:
50 = (x+2)/(1-x/(x+2))
Simplifying this expression, we get:
50 = (x+2)^2/(2x+2)
Multiplying both sides by 2x+2, we get:
50(2x+2) = (x+2)^2
Expanding the right side, we get:
50(2x+2) = x^2 + 4x + 4
Simplifying and rearranging, we get:
x^2 - 92x + 96 = 0
Solving for x using the quadratic formula, we get:
x = (92 ± √(92^2 - 4*96))/2
x ≈ 1.043 or x ≈ 90.957
Since the common ratio of a G.P must be between -1 and 1 for the sum to converge, we must have x ≈ 1.043.
Therefore, the first term is x+2 ≈ 3.043 and the common ratio is r ≈ 1.043/3.043 ≈ 0.342.
The G.P is 3.043, 1.043, 0.356, 0.122, 0.042, ...
The sum of the first 5 terms is 3.043+1.043+0.356+0.122+0.042 = 4.606.
Taking the limit as n goes to infinity, we see that the sum approaches 4.606/(1-0.342) = 6.975.
Therefore, the answer is (E) none of the above.
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If the first term of a G.P exceeds the second term by 2 and the sum to infinity is 50 the series is ________.a)10, 8, 32/5 …b)10, 8, 5/2…c)10, 10/3, 10/9…d)NoneCorrect answer is option 'A'. Can you explain this answer?
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