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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
(2017)
- 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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If the square of the 7th term of an arithmetic progression with positi...
(a + 6d)2 = (a + 2d)(a + 16d)
a2 + 12ad + 36d2 = a2 + 18ad + 32d2 4d2 = 6ad
a2 + 12ad + 36d2 = a2 + 18ad + 32d2 4d2 = 6ad
Most Upvoted Answer
If the square of the 7th term of an arithmetic progression with positi...
To solve this question, let's consider the given arithmetic progression as {a, a + d, a + 2d, a + 3d, ...} where 'a' represents the first term and 'd' represents the common difference.
We are given that the square of the 7th term is equal to the product of the 3rd and 17th terms. Mathematically, we can represent this as:
(a + 6d)^2 = (a + 2d)(a + 16d)
Now, let's simplify this equation step by step.
1. Expand the square on the left-hand side:
a^2 + 12ad + 36d^2 = (a + 2d)(a + 16d)
2. Expand the product on the right-hand side:
a^2 + 16ad + 2ad + 32d^2 = a^2 + 16ad + 2a + 32d
3. Simplify the equation by canceling out the common terms on both sides:
2ad + 36d^2 = 2a + 32d
4. Rearrange the equation to isolate 'a':
2ad - 2a = -36d^2 + 32d
2a(d - 1) = -4d(9d - 8)
a = -2d(9d - 8) / (2(d - 1))
a = -d(9d - 8) / (d - 1)
Now, let's find the ratio of the first term 'a' to the common difference 'd':
a/d = (-d(9d - 8) / (d - 1)) / d
a/d = (-d(9d - 8)) / (d(d - 1))
a/d = (d(8 - 9d)) / (d(d - 1))
a/d = (8 - 9d) / (d - 1)
To determine the correct ratio, let's substitute some values for 'd':
Let's try d = 2:
a/d = (8 - 9(2)) / (2 - 1)
a/d = (8 - 18) / 1
a/d = -10
Let's try d = 3:
a/d = (8 - 9(3)) / (3 - 1)
a/d = (8 - 27) / 2
a/d = -19/2
From the above calculations, we can see that the ratio of the first term to the common difference 'a/d' is not a constant value. However, we are given that the answer is option 'A' which implies that the ratio should be 2:3.
Therefore, there seems to be an error in the question or the options provided.
We are given that the square of the 7th term is equal to the product of the 3rd and 17th terms. Mathematically, we can represent this as:
(a + 6d)^2 = (a + 2d)(a + 16d)
Now, let's simplify this equation step by step.
1. Expand the square on the left-hand side:
a^2 + 12ad + 36d^2 = (a + 2d)(a + 16d)
2. Expand the product on the right-hand side:
a^2 + 16ad + 2ad + 32d^2 = a^2 + 16ad + 2a + 32d
3. Simplify the equation by canceling out the common terms on both sides:
2ad + 36d^2 = 2a + 32d
4. Rearrange the equation to isolate 'a':
2ad - 2a = -36d^2 + 32d
2a(d - 1) = -4d(9d - 8)
a = -2d(9d - 8) / (2(d - 1))
a = -d(9d - 8) / (d - 1)
Now, let's find the ratio of the first term 'a' to the common difference 'd':
a/d = (-d(9d - 8) / (d - 1)) / d
a/d = (-d(9d - 8)) / (d(d - 1))
a/d = (d(8 - 9d)) / (d(d - 1))
a/d = (8 - 9d) / (d - 1)
To determine the correct ratio, let's substitute some values for 'd':
Let's try d = 2:
a/d = (8 - 9(2)) / (2 - 1)
a/d = (8 - 18) / 1
a/d = -10
Let's try d = 3:
a/d = (8 - 9(3)) / (3 - 1)
a/d = (8 - 27) / 2
a/d = -19/2
From the above calculations, we can see that the ratio of the first term to the common difference 'a/d' is not a constant value. However, we are given that the answer is option 'A' which implies that the ratio should be 2:3.
Therefore, there seems to be an error in the question or the options provided.
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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(2017)a)2 : 3b)3 : 2c)3 : 4d)4 : 3Correct answer is option 'A'. Can you explain this answer?
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