(Probability: Recursion) Ant-problem

As I was working on this problem I figured I should be able to solve it by calculating the sum of the probability of the ant taking n steps to reach zero multiplied by n, which I believe is the following calculation (where i=1 represents the ant taking 5 steps, i=2:7 steps and so on): $\sum _{ i=1 }^{ \infty }{ { \left( \frac { 2 }{ 3 } \right) }^{ 4+i }{ \left( \frac { 1 }{ 3 } \right) }^{ i-1 } } {2i+1 \choose i-1} \left( 2i+3 \right)$

However, this yields too high of a result, and when I single out the probability-part of the calculation I get this: $\sum _{ i=1 }^{ \infty }{ { \left( \frac { 2 }{ 3 } \right) }^{ 4+i }{ \left( \frac { 1 }{ 3 } \right) }^{ i-1 } } {2i+1 \choose i-1}={ \left( \frac { 4 }{ 3 } \right) }$ which obviously can't be right since the total probability should add up to 1. What am I missing?

#Combinatorics

Easy Math Editor

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Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$