SOLUTION: SUM THIS SERIES: 1^3+(1^3+2^3)+(1^3+2^3+3^3)+.......(1^3+2^3+...50^3)

Question 1054887: SUM THIS SERIES:
1^3+(1^3+2^3)+(1^3+2^3+3^3)+.......(1^3+2^3+...50^3)
Answer by Edwin McCravy(20056) (Show Source): You can put this solution on YOUR website!
SUM THIS SERIES:
1^3+(1^3+2^3)+(1^3+2^3+3^3)+.......(1^3+2^3+...50^3)



We can find all the formulas we need here:

http://godplaysdice.blogspot.com/2008/12/how-could-you-guess-formula-for-sum-of.html

The nth term is the sum of the first n cubes of the positive
integers:

We look up the formula for the sum of the first n cubes of 
positive integers:



So we are looking for this:











Now we must look up the formula for the sum
of the first n 4th powers and squares of positive
integers:

The sum of the first n 4th powers of positive integers is 



The sum of the first n squares of powers of positive integers is 

sum of the first n squares
  

We have already looked up the formula for the sum of the first 
n cubes of powers of positive integers 
sum of the first n cubes, which again is  


 
Substituting for the summations



which simplifies to

 

which actually factors as



although that factorization isn't necessary

Substituting n = 50 gives

17240210

Edwin

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