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The first two terms of a geometric progression add up to 12. The sum of the third and the fourth terms is 48. If the terms of the geometric progression are alternately positive and negative, then the first term is
- a)4
- b)− 4
- c)12
- d)− 12
Correct answer is option 'D'. Can you explain this answer?
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Most Upvoted Answer
The first two terms of a geometric progression add up to 12. The sum o...
Let the GP be a, ar, ar2, ar3, ........arn−1
Where a = first term and r = Common ratio
According to question
We have t1+t2=12 ⇒a+ar=12 ...(i)
t3+t4=48 ⇒ ar2 + ar3 = 48 .....(ii)
Divide the equations (i) & (ii)
But the terms are alternately positive and negative,
∴ r = -2
Now using equation (i)
Where a = first term and r = Common ratio
According to question
We have t1+t2=12 ⇒a+ar=12 ...(i)
t3+t4=48 ⇒ ar2 + ar3 = 48 .....(ii)
Divide the equations (i) & (ii)
But the terms are alternately positive and negative,
∴ r = -2
Now using equation (i)
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The first two terms of a geometric progression add up to 12. The sum o...
Given information:
- First two terms add up to 12
- Sum of the third and fourth terms is 48
- Terms are alternately positive and negative
Solution:
Let the first term of the geometric progression be 'a' and the common ratio be 'r'.
First two terms sum up to 12:
Using the formula for the sum of the first two terms in a geometric progression:
a + ar = 12
Sum of the third and fourth terms is 48:
Using the formula for the sum of the first four terms in a geometric progression:
a + ar + ar^2 + ar^3 = 48
Terms are alternately positive and negative:
Since the terms are alternately positive and negative, the sign of 'r' will alternate between positive and negative.
Solving the equations:
From the first equation, we can express 'r' in terms of 'a':
a(1 + r) = 12
r = (12 - a)/a
Substitute this value of 'r' into the second equation and solve for 'a':
a + a(12 - a)/a + a(12 - a)^2/a^2 + a(12 - a)^3/a^3 = 48
a + 12 - a + 144 - 24a + a^2 + 144a^2 - 72a^2 + 12a - 48a + 4a^3 = 48
4a^3 - 36a^2 + 96a + 144 = 0
a^3 - 9a^2 + 24a + 36 = 0
(a - 12)(a^2 - 3a - 3) = 0
The possible values for 'a' are 12, (-3 + √21)/2, and (-3 - √21)/2. Since the terms are alternately positive and negative, the first term 'a' must be negative. Therefore, the first term is -12.
Therefore, the correct answer is option 'd' (-12).
- First two terms add up to 12
- Sum of the third and fourth terms is 48
- Terms are alternately positive and negative
Solution:
Let the first term of the geometric progression be 'a' and the common ratio be 'r'.
First two terms sum up to 12:
Using the formula for the sum of the first two terms in a geometric progression:
a + ar = 12
Sum of the third and fourth terms is 48:
Using the formula for the sum of the first four terms in a geometric progression:
a + ar + ar^2 + ar^3 = 48
Terms are alternately positive and negative:
Since the terms are alternately positive and negative, the sign of 'r' will alternate between positive and negative.
Solving the equations:
From the first equation, we can express 'r' in terms of 'a':
a(1 + r) = 12
r = (12 - a)/a
Substitute this value of 'r' into the second equation and solve for 'a':
a + a(12 - a)/a + a(12 - a)^2/a^2 + a(12 - a)^3/a^3 = 48
a + 12 - a + 144 - 24a + a^2 + 144a^2 - 72a^2 + 12a - 48a + 4a^3 = 48
4a^3 - 36a^2 + 96a + 144 = 0
a^3 - 9a^2 + 24a + 36 = 0
(a - 12)(a^2 - 3a - 3) = 0
The possible values for 'a' are 12, (-3 + √21)/2, and (-3 - √21)/2. Since the terms are alternately positive and negative, the first term 'a' must be negative. Therefore, the first term is -12.
Therefore, the correct answer is option 'd' (-12).
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The first two terms of a geometric progression add up to 12. The sum of the third and the fourth terms is 48. If the terms of the geometric progression are alternately positive and negative, then the first term isa)4b)− 4c)12d)− 12Correct answer is option 'D'. Can you explain this answer?
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The first two terms of a geometric progression add up to 12. The sum of the third and the fourth terms is 48. If the terms of the geometric progression are alternately positive and negative, then the first term isa)4b)− 4c)12d)− 12Correct answer is option 'D'. Can you explain this answer? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about The first two terms of a geometric progression add up to 12. The sum of the third and the fourth terms is 48. If the terms of the geometric progression are alternately positive and negative, then the first term isa)4b)− 4c)12d)− 12Correct answer is option 'D'. Can you explain this answer? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The first two terms of a geometric progression add up to 12. The sum of the third and the fourth terms is 48. If the terms of the geometric progression are alternately positive and negative, then the first term isa)4b)− 4c)12d)− 12Correct answer is option 'D'. Can you explain this answer?.
The first two terms of a geometric progression add up to 12. The sum of the third and the fourth terms is 48. If the terms of the geometric progression are alternately positive and negative, then the first term isa)4b)− 4c)12d)− 12Correct answer is option 'D'. Can you explain this answer? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about The first two terms of a geometric progression add up to 12. The sum of the third and the fourth terms is 48. If the terms of the geometric progression are alternately positive and negative, then the first term isa)4b)− 4c)12d)− 12Correct answer is option 'D'. Can you explain this answer? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The first two terms of a geometric progression add up to 12. The sum of the third and the fourth terms is 48. If the terms of the geometric progression are alternately positive and negative, then the first term isa)4b)− 4c)12d)− 12Correct answer is option 'D'. Can you explain this answer?.
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The first two terms of a geometric progression add up to 12. The sum of the third and the fourth terms is 48. If the terms of the geometric progression are alternately positive and negative, then the first term isa)4b)− 4c)12d)− 12Correct answer is option 'D'. Can you explain this answer?, a detailed solution for The first two terms of a geometric progression add up to 12. The sum of the third and the fourth terms is 48. If the terms of the geometric progression are alternately positive and negative, then the first term isa)4b)− 4c)12d)− 12Correct answer is option 'D'. Can you explain this answer? has been provided alongside types of The first two terms of a geometric progression add up to 12. The sum of the third and the fourth terms is 48. If the terms of the geometric progression are alternately positive and negative, then the first term isa)4b)− 4c)12d)− 12Correct answer is option 'D'. Can you explain this answer? theory, EduRev gives you an
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