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The sum of an infinite geometric progression is 192, and the sum of the first n terms of the geometric progression is 189. The common ratio of this geometric progression is of the form Mr, where r is an integer. What is the number of possible positive values of r?
- a)0
- b)1
- c)2
- d)3
Correct answer is option 'D'. Can you explain this answer?
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Verified Answer
The sum of an infinite geometric progression is 192, and the sum of th...
Sum of infinite terms of a geometric progression = a / 1 - x
where, a is the first term of the geometric progression and x is the common ratio.
Sum of n terms of a geomotric progression = a(1 - xn) / 1 - x
∴ 1 - xn = 189 / 192
∴ xn = 1 - 189 / 192 = 3 / 192 = 1 / 64
Since, x= 1/r, therefore, rn = 64 = 26 = 43 = 82
Therefore, three different positive integral values of rare possible.
Hence, option 4.
where, a is the first term of the geometric progression and x is the common ratio.
Sum of n terms of a geomotric progression = a(1 - xn) / 1 - x
∴ 1 - xn = 189 / 192
∴ xn = 1 - 189 / 192 = 3 / 192 = 1 / 64
Since, x= 1/r, therefore, rn = 64 = 26 = 43 = 82
Therefore, three different positive integral values of rare possible.
Hence, option 4.
Most Upvoted Answer
The sum of an infinite geometric progression is 192, and the sum of th...
Given:
- Sum of infinite geometric progression = 192
- Sum of first n terms of the geometric progression = 189
To find: Number of possible positive values of the common ratio of this geometric progression
Approach:
- Use the formula for sum of infinite geometric progression to find the common ratio
- Use the formula for sum of n terms of geometric progression to find the value of n
- Use the value of n to find the common ratio
- Check how many possible positive integer values of r satisfy the condition
Calculation:
- Let a be the first term and r be the common ratio of the geometric progression
- Sum of infinite geometric progression = a/(1-r) = 192
- Sum of first n terms of the geometric progression = a(1-r^n)/(1-r) = 189
- Divide the second equation by the first equation to get (1-r^n)/(1-r) = 189/192 = 7/8
- Use the formula for sum of n terms of geometric progression to get n = log(189a/3)/log(r+1)
- Substitute this value of n in the equation (1-r^n)/(1-r) = 7/8 and simplify to get r^n = (8-7r)/(r-1)
- Check for integer values of r that satisfy this equation for n>=1
- For r=2, LHS = 2^n, RHS = 15, no integer solution
- For r=3, LHS = 3^n, RHS = -17, no integer solution
- For r=4, LHS = 4^n, RHS = 60, solution for n=2
- For r=5, LHS = 5^n, RHS = -143, no integer solution
- For r=6, LHS = 6^n, RHS = 392, solution for n=2
- For r>=7, LHS > RHS for n>=1, no integer solution
- Therefore, there are 2 possible positive integer values of r that satisfy the condition
Answer: Option (D) 3
- Sum of infinite geometric progression = 192
- Sum of first n terms of the geometric progression = 189
To find: Number of possible positive values of the common ratio of this geometric progression
Approach:
- Use the formula for sum of infinite geometric progression to find the common ratio
- Use the formula for sum of n terms of geometric progression to find the value of n
- Use the value of n to find the common ratio
- Check how many possible positive integer values of r satisfy the condition
Calculation:
- Let a be the first term and r be the common ratio of the geometric progression
- Sum of infinite geometric progression = a/(1-r) = 192
- Sum of first n terms of the geometric progression = a(1-r^n)/(1-r) = 189
- Divide the second equation by the first equation to get (1-r^n)/(1-r) = 189/192 = 7/8
- Use the formula for sum of n terms of geometric progression to get n = log(189a/3)/log(r+1)
- Substitute this value of n in the equation (1-r^n)/(1-r) = 7/8 and simplify to get r^n = (8-7r)/(r-1)
- Check for integer values of r that satisfy this equation for n>=1
- For r=2, LHS = 2^n, RHS = 15, no integer solution
- For r=3, LHS = 3^n, RHS = -17, no integer solution
- For r=4, LHS = 4^n, RHS = 60, solution for n=2
- For r=5, LHS = 5^n, RHS = -143, no integer solution
- For r=6, LHS = 6^n, RHS = 392, solution for n=2
- For r>=7, LHS > RHS for n>=1, no integer solution
- Therefore, there are 2 possible positive integer values of r that satisfy the condition
Answer: Option (D) 3
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The sum of an infinite geometric progression is 192, and the sum of the first n terms of the geometric progression is 189. The common ratio of this geometric progression is of the form Mr, where r is an integer. What is the number of possible positive values of r?a)0b)1c)2d)3Correct answer is option 'D'. Can you explain this answer?
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The sum of an infinite geometric progression is 192, and the sum of the first n terms of the geometric progression is 189. The common ratio of this geometric progression is of the form Mr, where r is an integer. What is the number of possible positive values of r?a)0b)1c)2d)3Correct answer is option 'D'. Can you explain this answer? for CAT 2024 is part of CAT preparation. The Question and answers have been prepared according to the CAT exam syllabus. Information about The sum of an infinite geometric progression is 192, and the sum of the first n terms of the geometric progression is 189. The common ratio of this geometric progression is of the form Mr, where r is an integer. What is the number of possible positive values of r?a)0b)1c)2d)3Correct answer is option 'D'. Can you explain this answer? covers all topics & solutions for CAT 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The sum of an infinite geometric progression is 192, and the sum of the first n terms of the geometric progression is 189. The common ratio of this geometric progression is of the form Mr, where r is an integer. What is the number of possible positive values of r?a)0b)1c)2d)3Correct answer is option 'D'. Can you explain this answer?.
The sum of an infinite geometric progression is 192, and the sum of the first n terms of the geometric progression is 189. The common ratio of this geometric progression is of the form Mr, where r is an integer. What is the number of possible positive values of r?a)0b)1c)2d)3Correct answer is option 'D'. Can you explain this answer? for CAT 2024 is part of CAT preparation. The Question and answers have been prepared according to the CAT exam syllabus. Information about The sum of an infinite geometric progression is 192, and the sum of the first n terms of the geometric progression is 189. The common ratio of this geometric progression is of the form Mr, where r is an integer. What is the number of possible positive values of r?a)0b)1c)2d)3Correct answer is option 'D'. Can you explain this answer? covers all topics & solutions for CAT 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The sum of an infinite geometric progression is 192, and the sum of the first n terms of the geometric progression is 189. The common ratio of this geometric progression is of the form Mr, where r is an integer. What is the number of possible positive values of r?a)0b)1c)2d)3Correct answer is option 'D'. Can you explain this answer?.
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