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If Sn=2n3+n2+3n+1, where S_nS n denotes sum to first n terms of a series. It is given that tx , which is the xth term of the series is equal to 84, then x=?
- a)6
- b)5
- c)4
- d)7
Correct answer is option 'C'. Can you explain this answer?
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IfSn=2n3+n2+3n+1,where S_nS n denotes sum to first n terms of a serie...
We are given with Sn=2n3+n2+3n+1.
We know that any term tn can be found using tn=Sn−Sn−1.
Here, tx=Sx−Sx−1
So, 2x3+x2+3x+1−[2(x−1)3(x−1)2+3(x−1)+1]=84
⇒ 2x3 + x2 + 3x + 1 - [2 (x3 - 1 - 3x2 + 3x) + x2 + 1 - 2x + 3x — 3 + 1] =84
⇒ 2x3 + x2 + 3x + 1 - 2x3 + 2 + 6x2 - 6x - x2 - 1 + 2x - 3x + 3 - 1 = 84
⇒ 4 + 6x2 - 6x + 2x = 84
⇒ 6x2 - 4x = 80
⇒ 3x2 - 2x - 40 = 0
∴ x cannot be negative.
∴ x=4.
We know that any term tn can be found using tn=Sn−Sn−1.
Here, tx=Sx−Sx−1
So, 2x3+x2+3x+1−[2(x−1)3(x−1)2+3(x−1)+1]=84
⇒ 2x3 + x2 + 3x + 1 - [2 (x3 - 1 - 3x2 + 3x) + x2 + 1 - 2x + 3x — 3 + 1] =84
⇒ 2x3 + x2 + 3x + 1 - 2x3 + 2 + 6x2 - 6x - x2 - 1 + 2x - 3x + 3 - 1 = 84
⇒ 4 + 6x2 - 6x + 2x = 84
⇒ 6x2 - 4x = 80
⇒ 3x2 - 2x - 40 = 0
∴ x cannot be negative.
∴ x=4.
Most Upvoted Answer
IfSn=2n3+n2+3n+1,where S_nS n denotes sum to first n terms of a serie...
Given:
The sum of the first n terms of the series is given by Sn = 2n^3 + n^2 + 3n + 1.
To find:
The value of x if the xth term of the series is equal to 84.
Approach:
To find the value of x, we need to find the xth term of the series and equate it to 84. We can then solve for x.
Solution:
Step 1: Finding the general term of the series
To find the xth term of the series, we need to find the general term of the series. We can do this by subtracting the sum of the first (x-1) terms from the sum of the first x terms.
Let's find the sum of the first (x-1) terms:
S(x-1) = 2(x-1)^3 + (x-1)^2 + 3(x-1) + 1
Now, let's find the sum of the first x terms:
Sx = 2x^3 + x^2 + 3x + 1
The xth term (tx) is given by:
tx = Sx - S(x-1)
= [2x^3 + x^2 + 3x + 1] - [2(x-1)^3 + (x-1)^2 + 3(x-1) + 1]
= [2x^3 + x^2 + 3x + 1] - [2(x^3 - 3x^2 + 3x - 1) + (x^2 - 2x + 1) + 3x - 3 + 1]
= 84 (Given)
Step 2: Solve for x
Now, we can equate the expression for tx to 84 and solve for x.
84 = [2x^3 + x^2 + 3x + 1] - [2(x^3 - 3x^2 + 3x - 1) + (x^2 - 2x + 1) + 3x - 3 + 1]
Simplifying the equation:
84 = 2x^3 + x^2 + 3x + 1 - 2x^3 + 6x^2 - 6x + 2 + x^2 - 2x + 1 + 3x - 3 + 1
84 = 7x^2 + 3x + 2
Rearranging the equation:
7x^2 + 3x + 2 - 84 = 0
7x^2 + 3x - 82 = 0
Solving the quadratic equation:
x = (-b ± √(b^2 - 4ac)) / 2a
Plugging in the values:
x = (-(3) ± √((3)^2 - 4(7)(-82))) / (2(7))
x = (-3 ± √(9 + 2296)) / 14
x = (-3 ± √2305) / 14
Since x is a positive integer,
The sum of the first n terms of the series is given by Sn = 2n^3 + n^2 + 3n + 1.
To find:
The value of x if the xth term of the series is equal to 84.
Approach:
To find the value of x, we need to find the xth term of the series and equate it to 84. We can then solve for x.
Solution:
Step 1: Finding the general term of the series
To find the xth term of the series, we need to find the general term of the series. We can do this by subtracting the sum of the first (x-1) terms from the sum of the first x terms.
Let's find the sum of the first (x-1) terms:
S(x-1) = 2(x-1)^3 + (x-1)^2 + 3(x-1) + 1
Now, let's find the sum of the first x terms:
Sx = 2x^3 + x^2 + 3x + 1
The xth term (tx) is given by:
tx = Sx - S(x-1)
= [2x^3 + x^2 + 3x + 1] - [2(x-1)^3 + (x-1)^2 + 3(x-1) + 1]
= [2x^3 + x^2 + 3x + 1] - [2(x^3 - 3x^2 + 3x - 1) + (x^2 - 2x + 1) + 3x - 3 + 1]
= 84 (Given)
Step 2: Solve for x
Now, we can equate the expression for tx to 84 and solve for x.
84 = [2x^3 + x^2 + 3x + 1] - [2(x^3 - 3x^2 + 3x - 1) + (x^2 - 2x + 1) + 3x - 3 + 1]
Simplifying the equation:
84 = 2x^3 + x^2 + 3x + 1 - 2x^3 + 6x^2 - 6x + 2 + x^2 - 2x + 1 + 3x - 3 + 1
84 = 7x^2 + 3x + 2
Rearranging the equation:
7x^2 + 3x + 2 - 84 = 0
7x^2 + 3x - 82 = 0
Solving the quadratic equation:
x = (-b ± √(b^2 - 4ac)) / 2a
Plugging in the values:
x = (-(3) ± √((3)^2 - 4(7)(-82))) / (2(7))
x = (-3 ± √(9 + 2296)) / 14
x = (-3 ± √2305) / 14
Since x is a positive integer,
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IfSn=2n3+n2+3n+1,where S_nS n denotes sum to first n terms of a series. It is given that tx , which is the xth term of the series is equal to 84, then x=?a)6b)5c)4d)7Correct answer is option 'C'. Can you explain this answer?
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IfSn=2n3+n2+3n+1,where S_nS n denotes sum to first n terms of a series. It is given that tx , which is the xth term of the series is equal to 84, then x=?a)6b)5c)4d)7Correct answer is option 'C'. 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 IfSn=2n3+n2+3n+1,where S_nS n denotes sum to first n terms of a series. It is given that tx , which is the xth term of the series is equal to 84, then x=?a)6b)5c)4d)7Correct answer is option 'C'. Can you explain this answer? covers all topics & solutions for CAT 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for IfSn=2n3+n2+3n+1,where S_nS n denotes sum to first n terms of a series. It is given that tx , which is the xth term of the series is equal to 84, then x=?a)6b)5c)4d)7Correct answer is option 'C'. Can you explain this answer?.
IfSn=2n3+n2+3n+1,where S_nS n denotes sum to first n terms of a series. It is given that tx , which is the xth term of the series is equal to 84, then x=?a)6b)5c)4d)7Correct answer is option 'C'. 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 IfSn=2n3+n2+3n+1,where S_nS n denotes sum to first n terms of a series. It is given that tx , which is the xth term of the series is equal to 84, then x=?a)6b)5c)4d)7Correct answer is option 'C'. Can you explain this answer? covers all topics & solutions for CAT 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for IfSn=2n3+n2+3n+1,where S_nS n denotes sum to first n terms of a series. It is given that tx , which is the xth term of the series is equal to 84, then x=?a)6b)5c)4d)7Correct answer is option 'C'. Can you explain this answer?.
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