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The product of three consecutive terms of a G.P. is 512. If 4 is added to each of the first and the second of these terms, the three terms now from an A.P. Then the sum of the original three terms of the given G.P. is
- a)36
- b)24
- c)32
- d)28
Correct answer is option 'D'. Can you explain this answer?
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The product of three consecutive terms of a G.P. is 512. If 4 is added...
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The product of three consecutive terms of a G.P. is 512. If 4 is added...
Given:
- The product of three consecutive terms of a G.P. is 512.
- If 4 is added to each of the first and the second term of these terms, the three terms now form an A.P.
To find:
The sum of the original three terms of the given G.P.
Solution:
Let the three terms of the G.P. be a, ar, and ar^2.
According to the given information:
a * ar * ar^2 = 512 ... (1)
Adding 4 to the first and second terms:
(a + 4), (ar + 4), ar^2
Now, these three terms form an A.P.
The common difference (d) of an A.P. can be found by subtracting the second term from the first term or the third term from the second term.
Common difference (d) = (ar + 4) - (a + 4) = ar - a
The sum of the first and third terms (a + ar^2) is equal to twice the second term (2 * (ar + 4)) since it is an A.P.
(a + ar^2) = 2 * (ar + 4)
Simplifying, we get:
a + ar^2 = 2ar + 8 ... (2)
Now, we have two equations (1) and (2) with two variables (a and r).
From equation (1):
a * ar * ar^2 = 512
Taking cube root on both sides:
a * ar = 8 ... (3)
From equation (2):
a + ar^2 = 2ar + 8
Simplifying, we get:
a - 2ar + ar^2 = 8
Factoring, we get:
(a - ar)(1 - r) = 8
Using equation (3), substitute ar as 8:
(8 - 8r)(1 - r) = 8
Dividing both sides by 8:
(1 - r)(1 - 8r) = 1
Expanding, we get:
1 - 9r + 8r^2 = 1
Simplifying, we get:
8r^2 - 9r = 0
Factorizing, we get:
r(8r - 9) = 0
So, r = 0 or r = 9/8
If r = 0, then the terms of the G.P. will be 0, 0, 0, which is not possible since the product of the terms is 512.
Therefore, we consider r = 9/8.
Substituting r = 9/8 in equation (3), we get:
a * (9/8) = 8
Simplifying, we get:
a = 64/9
So, the three terms of the G.P. are:
a = 64/9, ar = (64/9) * (9/8) = 8, ar^2 = (64/9) * (9/8)^2 = 9
The sum of the three terms is:
64/9 + 8 + 9 = 64/9 + 72/9 + 81/9 = 217
- The product of three consecutive terms of a G.P. is 512.
- If 4 is added to each of the first and the second term of these terms, the three terms now form an A.P.
To find:
The sum of the original three terms of the given G.P.
Solution:
Let the three terms of the G.P. be a, ar, and ar^2.
According to the given information:
a * ar * ar^2 = 512 ... (1)
Adding 4 to the first and second terms:
(a + 4), (ar + 4), ar^2
Now, these three terms form an A.P.
The common difference (d) of an A.P. can be found by subtracting the second term from the first term or the third term from the second term.
Common difference (d) = (ar + 4) - (a + 4) = ar - a
The sum of the first and third terms (a + ar^2) is equal to twice the second term (2 * (ar + 4)) since it is an A.P.
(a + ar^2) = 2 * (ar + 4)
Simplifying, we get:
a + ar^2 = 2ar + 8 ... (2)
Now, we have two equations (1) and (2) with two variables (a and r).
From equation (1):
a * ar * ar^2 = 512
Taking cube root on both sides:
a * ar = 8 ... (3)
From equation (2):
a + ar^2 = 2ar + 8
Simplifying, we get:
a - 2ar + ar^2 = 8
Factoring, we get:
(a - ar)(1 - r) = 8
Using equation (3), substitute ar as 8:
(8 - 8r)(1 - r) = 8
Dividing both sides by 8:
(1 - r)(1 - 8r) = 1
Expanding, we get:
1 - 9r + 8r^2 = 1
Simplifying, we get:
8r^2 - 9r = 0
Factorizing, we get:
r(8r - 9) = 0
So, r = 0 or r = 9/8
If r = 0, then the terms of the G.P. will be 0, 0, 0, which is not possible since the product of the terms is 512.
Therefore, we consider r = 9/8.
Substituting r = 9/8 in equation (3), we get:
a * (9/8) = 8
Simplifying, we get:
a = 64/9
So, the three terms of the G.P. are:
a = 64/9, ar = (64/9) * (9/8) = 8, ar^2 = (64/9) * (9/8)^2 = 9
The sum of the three terms is:
64/9 + 8 + 9 = 64/9 + 72/9 + 81/9 = 217
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The product of three consecutive terms of a G.P. is 512. If 4 is added to each of the first and the second of these terms, the three terms now from an A.P. Then the sum of the original three terms of the given G.P. isa)36b)24c)32d)28Correct answer is option 'D'. Can you explain this answer?
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The product of three consecutive terms of a G.P. is 512. If 4 is added to each of the first and the second of these terms, the three terms now from an A.P. Then the sum of the original three terms of the given G.P. isa)36b)24c)32d)28Correct answer is option 'D'. Can you explain this answer? for JEE 2025 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about The product of three consecutive terms of a G.P. is 512. If 4 is added to each of the first and the second of these terms, the three terms now from an A.P. Then the sum of the original three terms of the given G.P. isa)36b)24c)32d)28Correct answer is option 'D'. Can you explain this answer? covers all topics & solutions for JEE 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The product of three consecutive terms of a G.P. is 512. If 4 is added to each of the first and the second of these terms, the three terms now from an A.P. Then the sum of the original three terms of the given G.P. isa)36b)24c)32d)28Correct answer is option 'D'. Can you explain this answer?.
The product of three consecutive terms of a G.P. is 512. If 4 is added to each of the first and the second of these terms, the three terms now from an A.P. Then the sum of the original three terms of the given G.P. isa)36b)24c)32d)28Correct answer is option 'D'. Can you explain this answer? for JEE 2025 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about The product of three consecutive terms of a G.P. is 512. If 4 is added to each of the first and the second of these terms, the three terms now from an A.P. Then the sum of the original three terms of the given G.P. isa)36b)24c)32d)28Correct answer is option 'D'. Can you explain this answer? covers all topics & solutions for JEE 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The product of three consecutive terms of a G.P. is 512. If 4 is added to each of the first and the second of these terms, the three terms now from an A.P. Then the sum of the original three terms of the given G.P. isa)36b)24c)32d)28Correct answer is option 'D'. Can you explain this answer?.
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The product of three consecutive terms of a G.P. is 512. If 4 is added to each of the first and the second of these terms, the three terms now from an A.P. Then the sum of the original three terms of the given G.P. isa)36b)24c)32d)28Correct answer is option 'D'. Can you explain this answer?, a detailed solution for The product of three consecutive terms of a G.P. is 512. If 4 is added to each of the first and the second of these terms, the three terms now from an A.P. Then the sum of the original three terms of the given G.P. isa)36b)24c)32d)28Correct answer is option 'D'. Can you explain this answer? has been provided alongside types of The product of three consecutive terms of a G.P. is 512. If 4 is added to each of the first and the second of these terms, the three terms now from an A.P. Then the sum of the original three terms of the given G.P. isa)36b)24c)32d)28Correct answer is option 'D'. Can you explain this answer? theory, EduRev gives you an
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