Interesting sum
( n = 1 ∑ 9 n ( n + 1 ) ( n + 2 ) ) − 2 9 0 = ?
The answer is 2680.
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1 solution
Nice solution sir.
Nice solution. Exactly how I expected the question to be solved
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What was the reason to subtract 290 from the summation ?
Nice Solution.
I have a slightly simpler solution,substitute n + 1 = x ,then you just have to find, x = 2 ∑ 1 0 x 3 − x ,which can be easily solved.I didn't get the answer as i thought that 290 was included in each term.Could you please put a bracket?
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the original question wasnt like this. i had written below the summation to subtract 290 from the answer.a moderator must have edited the question.
How did you came at 3rd step from 2nd step?? I mean how did you removed summation??
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He just substituted the values of the summations. If you were not aware of the closed form solutions of the sum of first n natural numbers, first n squares of natural numbers and first n cubes of natural numbers, I would recommend you to try to prove them yourself.
Just application of formulas. You should learn and memorize them here .
⎩ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎧ k = 1 ∑ n k = 2 n ( n + 1 ) k = 1 ∑ n k 2 = 6 n ( n + 1 ) ( 2 n + 1 ) k = 1 ∑ n k 3 = ( 2 n ( n + 1 ) ) 2
n = 1 ∑ 9 n ( n + 1 ) ( n + 2 ) = n = 1 ∑ 9 ( n 3 + 3 n 2 + 2 n ) = n = 1 ∑ 9 n 3 + 3 n = 1 ∑ 9 n 2 + 2 n = 1 ∑ 9 n = ( 2 9 ( 1 0 ) ) 2 + 3 ( 6 9 ( 1 0 ) ( 1 9 ) ) + 2 ( 2 9 ( 1 0 ) ) = 2 0 2 5 + 8 5 5 + 9 0 = 2 9 7 0
The required answer is 2 9 7 0 − 2 9 0 = 2 6 8 0 .