Consider an A.P. such that the product of its first term and the fourth term is 32 less than the product of the second and third term. If the sum of the first 4 terms of the same A.P. is equal to 12, then the sum of all the possible third terms of the A.P. is?Correct answer is '6'. Can you explain this answer?

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Consider an A.P. such that the product of its first term and the fourth term is 32 less than the product of the second and third term. If the sum of the first 4 terms of the same A.P. is equal to 12, then the sum of all the possible third terms of the A.P. is?

Correct answer is '6'. Can you explain this answer?

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Consider an A.P. such that the product of its first term and the fourt...

Let the first 4 terms of the A.P. be (a-3d), (a-d), (a+d) and (a+3d).
Sum of the 4 terms of the A.P.= 4a= 12
.'. a= 3.
It is given that (a-3d)(a+3d)=(a-d)(a+d)-32
⇒ a² - 9d²= a² - d² - 32
⇒ 8d² = 32
⇒ d² = 4
⇒ d = d = + 2.
When d=2, the first 4 terms are (3-6), (3-2), (3+2) and (3+6) which is -3,1,5 and 9.

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Consider an A.P. such that the product of its first term and the fourt...

Given Information:

- We have an arithmetic progression (A.P.).
- The product of the first term and the fourth term is 32 less than the product of the second and third term.
- The sum of the first 4 terms of the A.P. is equal to 12.

Solution:

Let's consider the terms of the A.P. as a, a + d, a + 2d, a + 3d.

Product of First and Fourth Term:

According to the given information, the product of the first term (a) and the fourth term (a + 3d) is 32 less than the product of the second term (a + d) and the third term (a + 2d).

Therefore, we can write the equation as:
a * (a + 3d) = (a + d) * (a + 2d) - 32

Expanding and simplifying the equation:
a² + 3ad = a² + 3ad + 2d² - 32

The a² and 3ad terms cancel out, leaving us with:
0 = 2d² - 32

Simplifying further:
2d² = 32
d² = 16
d = ±4

Sum of First 4 Terms:

The sum of the first 4 terms of the A.P. is given as 12.

Therefore, we can write the equation as:
a + (a + d) + (a + 2d) + (a + 3d) = 12

Simplifying the equation:
4a + 6d = 12

Substituting the value of d from the previous step:
4a + 6(±4) = 12
4a ± 24 = 12

Simplifying further:
4a = 12 ± 24
4a = 36 or 4a = -12

Solving for a in both cases:
a = 9 or a = -3

Finding the Third Term:

Now, we can find the third term of the A.P. using the values of a and d.

For a = 9, d = 4:
Third term = a + 2d = 9 + 2(4) = 9 + 8 = 17

For a = -3, d = -4:
Third term = a + 2d = -3 + 2(-4) = -3 - 8 = -11

Sum of All Possible Third Terms:

The possible third terms of the A.P. are 17 and -11.

Therefore, the sum of all the possible third terms is:
17 + (-11) = 6

Therefore, the sum of all the possible third terms of the A.P. is 6.

Answered by Wizius Careers · View profile

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Consider an A.P. such that the product of its first term and the fourth term is 32 less than the product of the second and third term. If the sum of the first 4 terms of the same A.P. is equal to 12, then the sum of all the possible third terms of the A.P. is?Correct answer is '6'. Can you explain this answer? for CAT 2024 is part of CAT preparation. The Question and answers have been prepared according to the CAT exam syllabus. Information about Consider an A.P. such that the product of its first term and the fourth term is 32 less than the product of the second and third term. If the sum of the first 4 terms of the same A.P. is equal to 12, then the sum of all the possible third terms of the A.P. is?Correct answer is '6'. Can you explain this answer? covers all topics & solutions for CAT 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Consider an A.P. such that the product of its first term and the fourth term is 32 less than the product of the second and third term. If the sum of the first 4 terms of the same A.P. is equal to 12, then the sum of all the possible third terms of the A.P. is?Correct answer is '6'. Can you explain this answer?.

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