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Find the closed form value for this sum: `sum_(i = 1)^n (i^2 + 2*i)`  

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svjr | Student, Undergraduate | (Level 2) Honors

Posted October 27, 2011 at 9:27 PM via web

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Find the closed form value for this sum: `sum_(i = 1)^n (i^2 + 2*i)`

 

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justaguide | College Teacher | (Level 2) Distinguished Educator

Posted October 27, 2011 at 10:44 PM (Answer #1)

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The value of `sum_(i=1)^n (i^2 + 2*i)` = `sum_(i = i)^n ( i^2) + sum_(i = 1)^n 2*i`

The sum of the squares of the first n natural numbers is given `n^3/3 + n^2/2 + n/6` and the sum of the first n natural numbers is `n^2/2 + n/2`

`sum_(i = 1)^n i^2 + sum_(i = 1)^n 2*i`

=> `n^3/3 + n^2/2 + n/6 + n^2/2 + n/2`

=> `n^3/3 + n^2 + (2n)/3`

The required value of the sum is `n^3/3 + n^2 + (2n)/3`

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