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 $\Large n\geq 1$

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 $\Large n\geq 1$

anonymous · Answer

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

anonymous · Answer

$$\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.

anonymous · Answer

thank you very much Mr Elias

anonymous · Answer

yw