Question

4. a) Show the formular
\(\Large \sum_{k=1}^{n}k^{3} = 1 + 2^{3} + 3^{3} +…+n^{3} = (\frac{n(n+1)}{2})^{2}\)
by induction over \(\Large n\geq 1\)

Answer 1

It is true for n=1.
Suppose it is true for n.
Prove that is true for n+1

Answer 2

\[\Large \sum_{k=1}^{n+1}k^{3} = 1 + 2^{3} + 3^{3} +…+n^{3} = \\(\frac{n(n+1)}{2})^{2}+(n+1)^3 =(n+1)^2\left ( \frac{ n^2} {2^2} +(n+1)\right)=\\
(n+1)^2\left ( \frac{ n^2+4n+4} {2^2}\right)=\\
(n+1)^2\left ( \frac{(n+2)^2} {2^2}\right)=
\]
We are done.

Answer 3

thank you very much Mr Elias

Answer 4

yw