Question

Can someone show me how to prove the summation formulas? And how we can prove these two.

Answer 1

i will try the first one

Answer 2

did you try induction

Answer 3

I have not tried induction since I am not aware of it.

Answer 4

1 base case let n=1 since it says \[n \ge 1\]
\[LHS=\sum 1^3=\frac{1(1+1)^2}{4}=1=RHS\]
true

Answer 5

then let \[\color{red}{n=k}\]
\[1^3+2^3+...+k^3=\frac{k^2(k+1)^2}{4}\]
\[\color{red}{n=k+1}\]
\[1^3+2^3+...+k^3+\color{red}{(k+1)^3}=\frac{k^2(k+1)^2}{4}+\color{red}{(k+1)^3}\]
\[RHS=\frac{k^2(k+1)^2+4(k+1)^3}{4}=\frac{(k+1)^2(k^2+4k+4 )}{4}\]
\[\huge \frac{(k+1)^2(k+2)^2}{4}\]
this finishes the proof since the last step is the same as adding k+1 to the RHS of the 1st statement

Answer 6

I'll give you the medal. However, doubt that's what the teacher is asking for. he taught us a different funky way . But the proofs you provided might help.

Answer 7

http://www.proofwiki.org/wiki/Sum_of_Sequence_of_Cubes

Answer 8

check this in