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The sum of all the terms of an infinite geometric progression is 30. The sum of the squares of all the terms of that progression is 300. What is the sum of the first two terms of the progression?
- a)15
- b)45/2
- c)15/2
- d)10
Correct answer is option 'B'. Can you explain this answer?
Verified Answer
The sum of all the terms of an infinite geometric progression is 30. T...
In an infinite geometric progression, if a is the first term and r the common ratio, then the sum of all the terms is
The series formed by the squares is also a G.P., with a2 as the first term and r2 as the common ratio, so the sum is
Dividing the second equation by the first, we get,
∴ r = 0.5 and a = 15
Hence, option 2.
This question is part of UPSC exam. View all CAT courses
Most Upvoted Answer
The sum of all the terms of an infinite geometric progression is 30. T...
To solve this problem, we need to use the formulas for the sum of an infinite geometric progression and the sum of the squares of an infinite geometric progression.
Let's denote the first term of the geometric progression as 'a' and the common ratio as 'r'.
Sum of an Infinite Geometric Progression:
The sum of an infinite geometric progression is given by the formula:
S = a / (1 - r)
Sum of Squares of an Infinite Geometric Progression:
The sum of the squares of an infinite geometric progression is given by the formula:
S^2 = (a^2) / (1 - r^2)
Given that the sum of all the terms is 30, we can write:
30 = a / (1 - r) ----(1)
And the sum of the squares of all the terms is 300, we can write:
300 = (a^2) / (1 - r^2) ----(2)
We can solve these two equations simultaneously to find the values of 'a' and 'r'.
First, let's rearrange equation (1) to solve for 'a':
a = 30(1 - r) ----(3)
Substitute this value of 'a' in equation (2):
300 = (30(1 - r)^2) / (1 - r^2)
Simplify and cross-multiply:
300(1 - r^2) = 30(1 - r)^2
Expand and simplify:
300 - 300r^2 = 30 - 60r + 30r^2
Rearrange and combine like terms:
30r^2 - 60r + 270 = 0
Divide the equation by 30 to simplify:
r^2 - 2r + 9 = 0
Now, we can solve this quadratic equation using the quadratic formula:
r = (-(-2) ± √((-2)^2 - 4(1)(9))) / (2(1))
Simplify the expression under the square root:
r = (2 ± √(4 - 36)) / 2
r = (2 ± √(-32)) / 2
Since the square root of a negative number is not defined in the real number system, there are no real solutions for 'r'.
Therefore, there is no valid geometric progression that satisfies the given conditions. The answer to this question is not valid.
Let's denote the first term of the geometric progression as 'a' and the common ratio as 'r'.
Sum of an Infinite Geometric Progression:
The sum of an infinite geometric progression is given by the formula:
S = a / (1 - r)
Sum of Squares of an Infinite Geometric Progression:
The sum of the squares of an infinite geometric progression is given by the formula:
S^2 = (a^2) / (1 - r^2)
Given that the sum of all the terms is 30, we can write:
30 = a / (1 - r) ----(1)
And the sum of the squares of all the terms is 300, we can write:
300 = (a^2) / (1 - r^2) ----(2)
We can solve these two equations simultaneously to find the values of 'a' and 'r'.
First, let's rearrange equation (1) to solve for 'a':
a = 30(1 - r) ----(3)
Substitute this value of 'a' in equation (2):
300 = (30(1 - r)^2) / (1 - r^2)
Simplify and cross-multiply:
300(1 - r^2) = 30(1 - r)^2
Expand and simplify:
300 - 300r^2 = 30 - 60r + 30r^2
Rearrange and combine like terms:
30r^2 - 60r + 270 = 0
Divide the equation by 30 to simplify:
r^2 - 2r + 9 = 0
Now, we can solve this quadratic equation using the quadratic formula:
r = (-(-2) ± √((-2)^2 - 4(1)(9))) / (2(1))
Simplify the expression under the square root:
r = (2 ± √(4 - 36)) / 2
r = (2 ± √(-32)) / 2
Since the square root of a negative number is not defined in the real number system, there are no real solutions for 'r'.
Therefore, there is no valid geometric progression that satisfies the given conditions. The answer to this question is not valid.
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The sum of all the terms of an infinite geometric progression is 30. The sum of the squares of all the terms of that progression is 300. What is the sum of the first two terms of the progression?a)15b)45/2c)15/2d)10Correct answer is option 'B'. Can you explain this answer?
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