I need help to solve this question :
Use induction to prove that, for n>=0:
3*5^0 + 3*5^1 + 3*5^2 + 3*5^3 + ...+ 3*5^n = 3*(5^(n+1)-1)/4
As 3 is a common factor on both sides it is sufficient to prove:
$\displaystyle 5^0 + 5^1 + 5^2 + 5^3 + ...+ 5^n = (5^{n+1}-1)/4$
To start we need to prove that base case (where $\displaystyle n=0$)
Then assume it true for $\displaystyle n=k$, then show that:
$\displaystyle 5^0 + 5^1 + 5^2 + 5^3 + ...+ 5^k = (5^{k+1}-1)/4$
implies that:
$\displaystyle 5^0 + 5^1 + 5^2 + 5^3 + ...+ 5^{k+1} = (5^{k+2}-1)/4$
You do this by adding $\displaystyle 5^{k+1}$ to both sides of the assumed case, that is:
$\displaystyle (5^0 + 5^1 + 5^2 + 5^3 + ...+ 5^k)+5^{k+1} = (5^{k+1}-1)/4+5^{k+1}$
and show that the right hand side is equal to $\displaystyle (5^{k+2}-1)/4$
CB