1. Proof by induction

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

2. Originally Posted by smh745
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

3. thanks a lot for your replay

I appreicated