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If Sn = n3 + n2 + n + 1 , where Sn denotes the sum of the first n terms of a series and tm = 291, then m is equal to?
- a)24
- b)30
- c)26
- d)20
Correct answer is option 'B'. Can you explain this answer?
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Most Upvoted Answer
If Sn= n3+ n2+ n + 1 , where Sndenotes the sum of the first n terms of...
To find the value of m in the given series, we need to first find a pattern in the series and then use the formula for the sum of the first n terms to solve for m.
Pattern in the series:
The given series is Sn = n^3 + n^2 + n + 1.
Let's calculate the first few terms of the series to identify a pattern:
When n = 1, Sn = 1^3 + 1^2 + 1 + 1 = 4
When n = 2, Sn = 2^3 + 2^2 + 2 + 1 = 15
When n = 3, Sn = 3^3 + 3^2 + 3 + 1 = 40
When n = 4, Sn = 4^3 + 4^2 + 4 + 1 = 85
When n = 5, Sn = 5^3 + 5^2 + 5 + 1 = 156
From these calculations, we can observe that the terms are increasing at an increasing rate. The difference between consecutive terms is also increasing.
Formula for the sum of the first n terms:
The sum of the first n terms of a series can be calculated using the formula Sn = (n/6)(2n^2 + 3n + 1).
Using the formula, we can solve for m:
291 = (m/6)(2m^2 + 3m + 1)
Let's simplify the equation:
291 = (2m^3 + 3m^2 + m)/6
Multiplying both sides by 6:
1746 = 2m^3 + 3m^2 + m
Rearranging the equation:
2m^3 + 3m^2 + m - 1746 = 0
Now, we need to find the value of m that satisfies this equation. We can use a numerical method such as trial and error or use a graphing calculator to find the value of m.
Using trial and error, we find that m = 30 satisfies the equation:
2(30)^3 + 3(30)^2 + 30 - 1746 = 0
Therefore, the value of m is 30. So, the correct answer is option B.
Pattern in the series:
The given series is Sn = n^3 + n^2 + n + 1.
Let's calculate the first few terms of the series to identify a pattern:
When n = 1, Sn = 1^3 + 1^2 + 1 + 1 = 4
When n = 2, Sn = 2^3 + 2^2 + 2 + 1 = 15
When n = 3, Sn = 3^3 + 3^2 + 3 + 1 = 40
When n = 4, Sn = 4^3 + 4^2 + 4 + 1 = 85
When n = 5, Sn = 5^3 + 5^2 + 5 + 1 = 156
From these calculations, we can observe that the terms are increasing at an increasing rate. The difference between consecutive terms is also increasing.
Formula for the sum of the first n terms:
The sum of the first n terms of a series can be calculated using the formula Sn = (n/6)(2n^2 + 3n + 1).
Using the formula, we can solve for m:
291 = (m/6)(2m^2 + 3m + 1)
Let's simplify the equation:
291 = (2m^3 + 3m^2 + m)/6
Multiplying both sides by 6:
1746 = 2m^3 + 3m^2 + m
Rearranging the equation:
2m^3 + 3m^2 + m - 1746 = 0
Now, we need to find the value of m that satisfies this equation. We can use a numerical method such as trial and error or use a graphing calculator to find the value of m.
Using trial and error, we find that m = 30 satisfies the equation:
2(30)^3 + 3(30)^2 + 30 - 1746 = 0
Therefore, the value of m is 30. So, the correct answer is option B.
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Community Answer
If Sn= n3+ n2+ n + 1 , where Sndenotes the sum of the first n terms of...
Sn - Sn = tn
Substitute m instead of n
Sm - Sm-1 = tm
We know that Sn = n3 + n2 + n + 1
Hence m3 + m2 + m + 1
m3 + m2 + m + 1 - [(m-1)3 + (m-1)2 + (m-1) + 1 ] = 291
m3 + m2 + m + 1 - [m3 - (3m)2 + 3m - 1 + m2 - 2m + 1 + m - 1 + 1] = 291
1 + (3m)2 + 3m + 1 -2m - 1 = 291
(-3m)2 + m - 290 = 0
(3m)2 - m + 290 = 0
Solving above equation we get m = -29, 30
M cannot be negative
Hence m = 30
Substitute m instead of n
Sm - Sm-1 = tm
We know that Sn = n3 + n2 + n + 1
Hence m3 + m2 + m + 1
m3 + m2 + m + 1 - [(m-1)3 + (m-1)2 + (m-1) + 1 ] = 291
m3 + m2 + m + 1 - [m3 - (3m)2 + 3m - 1 + m2 - 2m + 1 + m - 1 + 1] = 291
1 + (3m)2 + 3m + 1 -2m - 1 = 291
(-3m)2 + m - 290 = 0
(3m)2 - m + 290 = 0
Solving above equation we get m = -29, 30
M cannot be negative
Hence m = 30
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If Sn= n3+ n2+ n + 1 , where Sndenotes the sum of the first n terms of a series and tm= 291, then m is equal to?a)24b)30c)26d)20Correct answer is option 'B'. Can you explain this answer?
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