Base case, n=1:   LHS = 1^4 = 1

                  RHS = (1/30)(1*2*3*5) = (1/30)(30) = 1   (base case holds) 





Hypothesis:

Assume  + ... +  =    for  n=k





Step case:  Let n=k+1

Thus far, it has been setup.  The task now is to show LHS = RHS for n=k+1, and the proof will be complete.



LHS =  + ... +  



...which can also be written...



=  + ... +  +  



...apply the hypothesis to  terms...



=  +  





...expand and simplify (lots of steps omitted here)...

= 



Proof is complete, but lets show this last expression is of the form of the RHS  ...



Let u = k+1  -->  k=u-1:



LHS =  

    =    (= RHS)





 

Answer by robertb(5830)   (Show Source): You can put this solution on YOUR website!
 I will go directly to step 3 of the Inductive process, and will assume the Inductive hypothesis to be true, i.e., 



  for some n = k.



Prove:  is true for n = k + 1, that is, 



.





From the inductive hypothesis, 







Focus on the the polynomial  , which is of 4th degree.



By the factor theorem,  is a factor of this polynomial since .



Again by the factor theorem,  is a factor of this polynomial since .



In other words,  , where  is a quadratic expression.



Now    ===>     ===> .



Also,  is a 4th degree polynimial whose leading term is .  Since   also, this means .



===>  





===>  ===>   ===>  .



===> .





Therefore,  , 



and the proof is complete.



 

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