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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
- e)5 : 3
Correct answer is option 'A'. Can you explain this answer?
Most Upvoted Answer
If the square of the 7th term of an arithmetic progression with positi...
To solve this problem, let's first understand the given information and the properties of an arithmetic progression.
Given:
Square of the 7th term = Product of the 3rd and 17th terms
Let's assume the first term of the arithmetic progression is 'a' and the common difference is 'd'.
Properties of an arithmetic progression:
The nth term of an arithmetic progression can be represented as: a + (n-1)d
The square of the nth term can be represented as: (a + (n-1)d)^2
Now let's use the given information and the properties of an arithmetic progression to solve the problem.
The square of the 7th term = (a + 6d)^2
The product of the 3rd and 17th terms = (a + 2d)(a + 16d)
Since the square of the 7th term is equal to the product of the 3rd and 17th terms, we have:
(a + 6d)^2 = (a + 2d)(a + 16d)
Expanding both sides of the equation:
a^2 + 12ad + 36d^2 = a^2 + 18ad + 32d^2
Subtracting a^2 from both sides:
12ad + 36d^2 = 18ad + 32d^2
Rearranging the terms:
6ad = -4d^2
Dividing both sides by 2d:
3a = -2d
Dividing both sides by -2:
a/d = -2/3
Since the common difference 'd' is positive, we can rewrite the ratio as:
a/d = 2/(-3)
Simplifying the ratio:
a/d = -2/3
Therefore, the ratio of the first term to the common difference is 2 : 3, which corresponds to option A.
Hence, the correct answer is option A) 2 : 3.
Given:
Square of the 7th term = Product of the 3rd and 17th terms
Let's assume the first term of the arithmetic progression is 'a' and the common difference is 'd'.
Properties of an arithmetic progression:
The nth term of an arithmetic progression can be represented as: a + (n-1)d
The square of the nth term can be represented as: (a + (n-1)d)^2
Now let's use the given information and the properties of an arithmetic progression to solve the problem.
The square of the 7th term = (a + 6d)^2
The product of the 3rd and 17th terms = (a + 2d)(a + 16d)
Since the square of the 7th term is equal to the product of the 3rd and 17th terms, we have:
(a + 6d)^2 = (a + 2d)(a + 16d)
Expanding both sides of the equation:
a^2 + 12ad + 36d^2 = a^2 + 18ad + 32d^2
Subtracting a^2 from both sides:
12ad + 36d^2 = 18ad + 32d^2
Rearranging the terms:
6ad = -4d^2
Dividing both sides by 2d:
3a = -2d
Dividing both sides by -2:
a/d = -2/3
Since the common difference 'd' is positive, we can rewrite the ratio as:
a/d = 2/(-3)
Simplifying the ratio:
a/d = -2/3
Therefore, the ratio of the first term to the common difference is 2 : 3, which corresponds to option A.
Hence, the correct answer is option A) 2 : 3.
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If the square of the 7th term of an arithmetic progression with positi...
To solve this problem, let's denote the first term of the arithmetic progression as 'a' and the common difference as 'd'.
The 7th term of the arithmetic progression can be represented as: a + 6d
According to the given information, the square of the 7th term is equal to the product of the 3rd and 17th terms: (a + 6d)2 = (a + 2d)(a + 16d)
Expanding the left side of the equation: a2 + 12ad + 36d2 = a2 + 18ad + 32d2
Simplifying the equation: 6ad + 4d2 = 0
Factoring out '2d': 2d(3a + 2d) = 0
Since the common difference 'd' cannot be zero (as it is positive), we can conclude that: 3a + 2d = 0
Rearranging the equation to solve for 'a': 3a = -2d a = -2d/3
The ratio of the first term to the common difference is given by a/d: a/d = (-2d/3) / d a/d = -2/3
Therefore, the ratio of the first term to the common difference is 2:3, which corresponds to option A.
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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 : 3b)3 : 2c)3 : 4d)4 : 3e)5 : 3Correct answer is option 'A'. Can you explain this answer?
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