Question

I need verification for a problem of mathematical induction

Answer 1

\[\sum_{i=1}^{n} \frac{ 1 }{3{^i} }= \frac{ 1 }{ 2 }(1- \frac{1}{3{^n}}) for n \ge 1\]

Answer 2

Basis: If n=1: \[\sum_{i=1}^{1} \frac{1}{3{^1}} = \frac{1}{2} (1-\frac{1}{3{^1}}) \]
LHS = \[\frac{1}{3{^1}} = \frac{1}{3} \]RHS = \[\frac{1}{2}(\frac{1}{3})=\frac{2}{6}= \frac{1}{3} \] So the basis is true.

Answer 3

Assume that \[ \sum_{i}^{k} \frac{1}3{^i} = \frac{1}{2}(1-\frac{1}{3{^k}})\] is true

Answer 4

Prove for k+1. I have a lot of work but I'll wait for someone to volunteer to hlp so I don't waste time typing it out for no one.