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If the square of the 7th term of an arithmetic progression with positive common difference equals the product of the 3rd and 17th terms, then the ratio of the first term to the common difference is
- a)2:3
- b)3:2
- c)3:4
- d)4:3
Correct answer is option 'A'. Can you explain this answer?
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Verified Answer
If the square of the 7th term of an arithmetic progression with positi...
(a+6d)2 = (a+2d)(a+16d)
a2 + 12 ad + 36d2 = a2+ 18 ad + 32d2
Since, d is positive,
We get the ratio of a:d = 2:3
Option (A)
a2 + 12 ad + 36d2 = a2+ 18 ad + 32d2
Since, d is positive,
We get the ratio of a:d = 2:3
Option (A)
Most Upvoted Answer
If the square of the 7th term of an arithmetic progression with positi...
To solve this problem, we need to use the given information to form equations and then solve them to find the ratio of the first term to the common difference. Let's break down the solution into steps:
Step 1: Understanding the given information
We are given that the square of the 7th term of an arithmetic progression (AP) with positive common difference is equal to the product of the 3rd and 17th terms.
Step 2: Formulating equations
Let's assume that the first term of the AP is 'a' and the common difference is 'd'. We can then write the following equations based on the given information:
- The 7th term of the AP: a + 6d (since the 7th term is obtained by adding 6 times the common difference to the first term)
- The square of the 7th term: (a + 6d)^2
- The 3rd term of the AP: a + 2d (since the 3rd term is obtained by adding 2 times the common difference to the first term)
- The 17th term of the AP: a + 16d (since the 17th term is obtained by adding 16 times the common difference to the first term)
According to the given information, we have the equation: (a + 6d)^2 = (a + 2d)(a + 16d)
Step 3: Solving the equation
Expanding the equation and simplifying, we get:
a^2 + 12ad + 36d^2 = a^2 + 18ad + 32d^2
Cancelling out the common terms and rearranging, we get:
6ad = -4d^2
Dividing both sides by 2d (since d cannot be zero as given in the question), we get:
3a = -2d
Rearranging the equation, we find:
a/d = -2/3
Step 4: Finding the ratio of the first term to the common difference
Since a/d = -2/3, the ratio of the first term to the common difference is 2:3.
Therefore, the correct answer is option A) 2:3.
Step 1: Understanding the given information
We are given that the square of the 7th term of an arithmetic progression (AP) with positive common difference is equal to the product of the 3rd and 17th terms.
Step 2: Formulating equations
Let's assume that the first term of the AP is 'a' and the common difference is 'd'. We can then write the following equations based on the given information:
- The 7th term of the AP: a + 6d (since the 7th term is obtained by adding 6 times the common difference to the first term)
- The square of the 7th term: (a + 6d)^2
- The 3rd term of the AP: a + 2d (since the 3rd term is obtained by adding 2 times the common difference to the first term)
- The 17th term of the AP: a + 16d (since the 17th term is obtained by adding 16 times the common difference to the first term)
According to the given information, we have the equation: (a + 6d)^2 = (a + 2d)(a + 16d)
Step 3: Solving the equation
Expanding the equation and simplifying, we get:
a^2 + 12ad + 36d^2 = a^2 + 18ad + 32d^2
Cancelling out the common terms and rearranging, we get:
6ad = -4d^2
Dividing both sides by 2d (since d cannot be zero as given in the question), we get:
3a = -2d
Rearranging the equation, we find:
a/d = -2/3
Step 4: Finding the ratio of the first term to the common difference
Since a/d = -2/3, the ratio of the first term to the common difference is 2:3.
Therefore, the correct answer is option A) 2:3.
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Community Answer
If the square of the 7th term of an arithmetic progression with positi...
A
(a+6d)² =(a+2d)*(a+16d)
solve this equation you will get a:d=2:3
(a+6d)² =(a+2d)*(a+16d)
solve this equation you will get a:d=2:3
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If the square of the 7th term of an arithmetic progression with positive common difference equals the product of the 3rd and 17th terms, then the ratio of the first term to the common difference isa)2:3b)3:2c)3:4d)4:3Correct answer is option 'A'. Can you explain this answer?
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If the square of the 7th term of an arithmetic progression with positive common difference equals the product of the 3rd and 17th terms, then the ratio of the first term to the common difference isa)2:3b)3:2c)3:4d)4:3Correct answer is option 'A'. 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 If the square of the 7th term of an arithmetic progression with positive common difference equals the product of the 3rd and 17th terms, then the ratio of the first term to the common difference isa)2:3b)3:2c)3:4d)4:3Correct answer is option 'A'. Can you explain this answer? covers all topics & solutions for CAT 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If the square of the 7th term of an arithmetic progression with positive common difference equals the product of the 3rd and 17th terms, then the ratio of the first term to the common difference isa)2:3b)3:2c)3:4d)4:3Correct answer is option 'A'. Can you explain this answer?.
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