Introduction to Principle of mathematical induction, explanation using dominos

Proof of Arithmetic Progression using PMI
a+(a+d)+(a+2d)+…+(a+(n-1)d) = \frac{n}{2} (2a+(n-1)d)

Question 1.  1+3+3^2+…+3^{n-1} = \frac{(3^n-1)}{2}

Question 2. 1^3+2^3+3^3+…+n^3= \left ( \frac{n(n+1)}{2} \right )^2

Question 3. 1+\frac{1}{(1+2)}+\frac{1}{(1+2+3)}+…+\frac{1}{(1+2+3+…+n)}=\frac{2n}{(n+1)}

Question 4. 1.2.3+2.3.4+…+n(n+1)(n+2)=\frac{n(n+1)(n+2)(n+3)}{4}

Question 5. 1.3+2.3^2+3.3^3+…+n.3^n=\frac{(2n-1)3^{n+1}+3}{4}