You wish to prove $\displaystyle \displaystyle \begin{align*} 1 \cdot 1! + 2\cdot 2! + 3\cdot 3! + \dots + n \cdot n! = (n + 1)! - 1 \end{align*}$.
Base step, where $\displaystyle \displaystyle \begin{align*} n = 1 \end{align*}$:
$\displaystyle \displaystyle \begin{align*} LHS &= 1 \cdot 1! \\ &= 1 \\ \\ RHS &= (1 + 1)! - 1 \\ &= 1 \\ &= LHS \end{align*}$
Inductive step, assume the statement $\displaystyle \displaystyle \begin{align*} 1 \cdot 1! + 2\cdot 2! + 3 \cdot 3! + \dots + n \cdot n! = (n + 1)! - 1 \end{align*}$ holds true for $\displaystyle \displaystyle \begin{align*} n = k \end{align*}$, and use this to show it holds true for $\displaystyle \displaystyle \begin{align*} n = k + 1 \end{align*}$, i.e. show that $\displaystyle \displaystyle \begin{align*} 1 \cdot 1! + 2\cdot 2! + 3\cdot 3! + \dots + k\cdot k! + (k + 1)\cdot (k + 1)! = (k + 2)! - 1 \end{align*}$.
When we let $\displaystyle \displaystyle \begin{align*} n = k + 1 \end{align*}$ we have
$\displaystyle \displaystyle \begin{align*} LHS &= 1 \cdot 1! + 2 \cdot 2! + 3\cdot 3! + \dots + k \cdot k! + (k + 1)\cdot (k + 1)! \\ &= (k + 1)! - 1 + (k + 1)\cdot (k + 1)! \\ &= (k + 1)! + (k + 1)(k + 1)! - 1 \\ &= (k + 1)!( 1 + k + 1 ) - 1 \\ &= (k + 1)!( k + 2) - 1 \\ &= (k + 2)! - 1 \\ &= RHS \end{align*}$
Q.E.D.
For the first one,
Base case: $\displaystyle 1\times1! = 1 = (1+1)! - 1.$
Now, assume that the statement is true for $\displaystyle n = k.$ So, we have
$\displaystyle 1\times1! + 2\times2! + 3\times3! +\cdots+ k\times k! = (k+1)! - 1$
$\displaystyle \Rightarrow1\times1! + 2\times2! + 3\times3! +\cdots+ k\times k! + (k+1)(k+1)!$
. . . .$\displaystyle = (k+1)! - 1 + (k+1)(k+1)!$
$\displaystyle \Rightarrow1\times1! + 2\times2! + 3\times3! +\cdots+ k\times k! + (k+1)(k+1)!$
. . . .$\displaystyle = (k+1)! + (k+1)(k+1)! - 1$
. . . .$\displaystyle = (k+1)!\left[1 + (k+1)\right] - 1$
. . . .$\displaystyle = (k+1)!(k+2) - 1$
. . . .$\displaystyle = (k+2)! - 1$
Proof to part b:
Base: Where n=1:
LHS=1/2!
RHS=((1+1)!-1)/(1+1)!=(2!-1)/2!=1/2!=LHS
Now we assume the statement holds true for n=k, and show that it holds true for n=k+1
So we have
1/2!+2/3!+3/4!+4/5!+⋯+k/(k+1)!=((k+1)!-1)/(k+1)!
And want to show that
1/2!+2/3!+3/4!+4/5!+⋯+k/(k+1)!+((k+1))/(k+2)!=((k+2)!-1)/(k+2)!
LHS=1/2!+2/3!+3/4!+4/5!+⋯+k/(k+1)!+((k+1))/(k+2)!
=((k+1)!-1)/(k+1)!+((k+1))/(k+2)!
=((k+2)!(k+1)!-(k+2)!+(k+1)(k+1)!)/(k+1)!(k+2)!
=((k+2)!(k+1)!-(k+2)(k+1)!+(k+1)(k+1)!)/(k+1)!(k+2)!
=((k+2)!-(k+2)+(k+1))/(k+2)!
=((k+2)!-k+k-2+1)/(k+2)!
=((k+2)!-1)/(k+2)!
Q.E.D.