Question

Prove each of the following for all n ≥ 1 by the Principle of Mathematical Induction.

∑[i=1,n]i^3=(n^2(n+1)^2/(4)=(∑[i=1,n,i])^2

Answer 1

this is a well known sum
too hard to write the proof here, but if you google it you will find many, including ones n youtube

Answer 2

yeah I have looked and can not find anything could you try writing it out for me on here

Answer 3

http://www.youtube.com/watch?v=ZtvCVgShG88

Answer 4

case \(n=1\) as usual is trivial

Answer 5

then it is algebra you get to assume
\[1^2+2^3+3^3+...+k^3=\left(\frac{k(k+1)}{2}\right)^2\]

Answer 6

then you need to show
\[1^3+2^3+3^3+...+k^3+(k+1)^3=\left(\frac{(k+1)(k+2)}{2}\right)^2\]

Answer 7

this boils down to showing
\[\left(\frac{k(k+1)}{2}\right)^2+(k+1)^2=\left(\frac{(k+1)(k+2)}{2}\right)^2\]

Answer 8

damn \[\left(\frac{k(k+1)}{2}\right)^2+(k+1)^3=\left(\frac{(k+1)(k+2)}{2}\right)^2\]

Answer 9

and that is a raft of elementary algebra

Answer 10

so that is the answer

Answer 11

to the "prove" problem, you don't have answer, just the process how to get what they want you to prove. for this, you start at hypothesis step ( true when n =k) then add both sides of it by (k+1) ^2 , then, make some steps to show the final is the form of the case of (k+1)