Find an expression for the sum of the series

AI Thread Summary
The series in question is expressed as (1x2x6) + (2x3x7) + ... + n(n + 1)(n + 5). To find the sum, it is suggested that the result will be a polynomial of degree four, since the terms are cubic. The general form of the polynomial is proposed as S = a*n^4 + b*n^3 + c*n^2 + d*n + e. By evaluating the series for n = 1 to 5, a system of equations can be established to solve for the coefficients a, b, c, d, and e. This method provides a structured approach to derive the expression for the sum of the series.
MegaDeth
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Homework Statement



(1x2x6) + (2x3x7) + ... + n(n + 1)(n + 5)



Homework Equations



Find an expression for the sum of the series. Give your answer as a product of linear factors in n.


The Attempt at a Solution



I haven't tried it since I don't know what to do.
 
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I haven't seen one of those for years! They are interesting. There is some theory saying that the sum is equal to a polynomial with one degree higher than the term. Your terms are 3rd degree, so you need a 4th degree polynomial:
S = a*n^4 + b*n^3 + c*n^2 + d*n + e
By hand you figure out the sum and the powers for n = 1, 2, 3, 4, and 5.
That gives you a system of 5 equations with 5 unknowns a, b, c, d, e.
Solve the system to get the unknown coefficients.
 
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Thread 'Variable mass system : water sprayed into a moving container'

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