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The first term of an AP is equal to the sum of the common ratio of a GP and the first term of the GP, which is equal to the common difference of the AP. If the sum of the first two terms of the GP is equal to the sum of the first 2 terms of the AP, then the ratio of the first term of the GP to the first term of the AP is
- a)> 1
- b)independent of the first term of the GP
- c)< />
- d)independent of the first term of the AP
Correct answer is option 'B'. Can you explain this answer?
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Most Upvoted Answer
The first term of an AP is equal to the sum of the common ratio of a ...
We know that the three term of the AP are α - β, α, α + β and those of the GP are a, ar, ar2.
So, according to the quesition,
α - β = r + a = β
a + ar = 2α - β
a + ar = 2α - 2β + β
a+ar = 2
a(1 + r) = 2 (r + a) + r + a
a(1+ r) = 3 (r + a)
a + r = a(1 + r)/3
Now, the required ratio is a/α - β = a/(a + r) = (a × 3)/(a (1 + r)) = 3/(1 + r)
This is independent of the first term of the GP.
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Community Answer
The first term of an AP is equal to the sum of the common ratio of a ...
Explanation:
Given Information:
- First term of AP = common ratio of GP + first term of GP = common difference of AP
- Sum of first two terms of GP = sum of first two terms of AP
Calculating the terms:
Let the first term of the AP be 'a' and the common difference be 'd'.
Let the first term of the GP be 'b' and the common ratio be 'r'.
Given:
a = r + b
d = r
The sum of the first two terms of the AP is: a + (a + d) = 2a + d
The sum of the first two terms of the GP is: b + br = b(1 + r)
Given:
2a + d = b(1 + r)
Substitute the values of a, d, and r:
2(r + b) + r = b(1 + r)
2r + 2b + r = b + br
3r + 2b = b(1 + r)
3r + 2b = b + br
3r = b
Ratio of first terms:
Therefore, the ratio of the first term of the GP to the first term of the AP is:
b/a = 3r/(r + b) = 3/(1 + b/r)
Since b and r are independent of each other, the ratio b/a is independent of the first term of the GP. Hence, the correct answer is option 'B'.
Given Information:
- First term of AP = common ratio of GP + first term of GP = common difference of AP
- Sum of first two terms of GP = sum of first two terms of AP
Calculating the terms:
Let the first term of the AP be 'a' and the common difference be 'd'.
Let the first term of the GP be 'b' and the common ratio be 'r'.
Given:
a = r + b
d = r
The sum of the first two terms of the AP is: a + (a + d) = 2a + d
The sum of the first two terms of the GP is: b + br = b(1 + r)
Given:
2a + d = b(1 + r)
Substitute the values of a, d, and r:
2(r + b) + r = b(1 + r)
2r + 2b + r = b + br
3r + 2b = b(1 + r)
3r + 2b = b + br
3r = b
Ratio of first terms:
Therefore, the ratio of the first term of the GP to the first term of the AP is:
b/a = 3r/(r + b) = 3/(1 + b/r)
Since b and r are independent of each other, the ratio b/a is independent of the first term of the GP. Hence, the correct answer is option 'B'.
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The first term of an AP is equal to the sum of the common ratio of a GP and the first term of the GP, which is equal to the common difference of the AP. If the sum of the first two terms of the GP is equal to the sum of the first 2 terms of the AP, then the ratio of the first term of the GP to the first term of the AP isa)> 1b)independent of the first term of the GPc)d)independent of the first term of the APCorrect answer is option 'B'. Can you explain this answer?
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The first term of an AP is equal to the sum of the common ratio of a GP and the first term of the GP, which is equal to the common difference of the AP. If the sum of the first two terms of the GP is equal to the sum of the first 2 terms of the AP, then the ratio of the first term of the GP to the first term of the AP isa)> 1b)independent of the first term of the GPc)d)independent of the first term of the APCorrect answer is option 'B'. Can you explain this answer? for Electronics and Communication Engineering (ECE) 2024 is part of Electronics and Communication Engineering (ECE) preparation. The Question and answers have been prepared according to the Electronics and Communication Engineering (ECE) exam syllabus. Information about The first term of an AP is equal to the sum of the common ratio of a GP and the first term of the GP, which is equal to the common difference of the AP. If the sum of the first two terms of the GP is equal to the sum of the first 2 terms of the AP, then the ratio of the first term of the GP to the first term of the AP isa)> 1b)independent of the first term of the GPc)d)independent of the first term of the APCorrect answer is option 'B'. Can you explain this answer? covers all topics & solutions for Electronics and Communication Engineering (ECE) 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The first term of an AP is equal to the sum of the common ratio of a GP and the first term of the GP, which is equal to the common difference of the AP. If the sum of the first two terms of the GP is equal to the sum of the first 2 terms of the AP, then the ratio of the first term of the GP to the first term of the AP isa)> 1b)independent of the first term of the GPc)d)independent of the first term of the APCorrect answer is option 'B'. Can you explain this answer?.
The first term of an AP is equal to the sum of the common ratio of a GP and the first term of the GP, which is equal to the common difference of the AP. If the sum of the first two terms of the GP is equal to the sum of the first 2 terms of the AP, then the ratio of the first term of the GP to the first term of the AP isa)> 1b)independent of the first term of the GPc)d)independent of the first term of the APCorrect answer is option 'B'. Can you explain this answer? for Electronics and Communication Engineering (ECE) 2024 is part of Electronics and Communication Engineering (ECE) preparation. The Question and answers have been prepared according to the Electronics and Communication Engineering (ECE) exam syllabus. Information about The first term of an AP is equal to the sum of the common ratio of a GP and the first term of the GP, which is equal to the common difference of the AP. If the sum of the first two terms of the GP is equal to the sum of the first 2 terms of the AP, then the ratio of the first term of the GP to the first term of the AP isa)> 1b)independent of the first term of the GPc)d)independent of the first term of the APCorrect answer is option 'B'. Can you explain this answer? covers all topics & solutions for Electronics and Communication Engineering (ECE) 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The first term of an AP is equal to the sum of the common ratio of a GP and the first term of the GP, which is equal to the common difference of the AP. If the sum of the first two terms of the GP is equal to the sum of the first 2 terms of the AP, then the ratio of the first term of the GP to the first term of the AP isa)> 1b)independent of the first term of the GPc)d)independent of the first term of the APCorrect answer is option 'B'. Can you explain this answer?.
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