Understanding Mathematical Induction problems I have problem to understand this 3 formulas, I am new in this type of problems.
I have to solve this problems by induction.
\begin{align}
\sum_{j = 1}^{j = n} j^3 &= \left(\frac{n(n + 1)}{2}\right) ^ 2  &\text{where } n \geq 1 \\
\sum_{j = 1}^{j = n} j(j + 1) &= \frac{1}{3}n(n + 1)(n + 2)  & \text{where } n \geq 1 \\
\sum_{j = 1}^{j = n} j(j!) &= (n + 1)! - 1 
\end{align}

Here is an example of using induction that doesn't use sigma notation.
The OP can put it in their #Education Reference Folder.
Show that
$$\tag 1 1 + 2 + \dots + n = \frac{n(n+1)}{2}$$
Base Case:
True when $n = 1$ since $1 = \frac{1(1+1)}{2}$.
Inductive Step:
Assume that $1 + 2 + \dots + k = \frac{k(k+1)}{2}$.
Then $1 + 2 + \dots + k + (k+1) = (1 + 2 + \dots + k) + (k+1) =\frac{k(k+1)}{2} + (k+1) =$
$\quad (k+1) (\frac{k}{2} + 1) =\frac{(k+1)(k+2)}{2}$
So by induction, $\text{(1)}$ is always true.
