I'm not too sure where I'm going wrong. Any help would be appreciated.
Q. Prove $(X_{n} - \mu*n)^2 +\sigma^2n$ is a martingale
Set $X_{n+1} = X_{n} + Y_{n+1}$ as a Random Walk with mean $E[Y_{i}]=\mu$ and variance $Var(Y_{i})=\sigma^2$
My working: (Proving integrability is fine)
$E[(X_{n+1} - \mu*(n+1))^2|F_{n}]$
$E[(X_{n} + Y_{n+1} - \mu*n-\mu)^2|F_{n}]$
$E[(X_{n})^2|F_{n}]$ + $E[2*X_{n}*Y_{n+1}|F_{n}]$ - $E[2*X_{n}*\mu*n|F_{n}]$ - $E[2*X_{n}*\mu|F_{n}]$ + $E[Y_{n+1}^2|F_{n}]$ - $E[Y_{n+1}*\mu*n|F_{n}]$ - $E[Y_{n+1}*\mu|F_{n}]$ + $E[(\mu*n)^2|F_{n}]$ + $E[\mu^2*n|F_{n}]$ + $E[\mu*^2|F_{n}]$
$(X_{n})^2 + (2*X_{n}*\mu) - (2*X_{n}*\mu*n) - (2*X_{n}*\mu) - \sigma^2 - (2\mu^2*n) - (2\mu^2) + (\mu*n)^2 + 2*\mu^2*n + \mu^2$
$(X_{n})^2 - (2*X_{n}*\mu*n) - \sigma^2 + (\mu*n)^2 - (\mu^2) $
Add the $\sigma^2(n+1)$ from the end of the question.
$(X_{n} - (\mu*n))^2 - \sigma^2 + \sigma^2(n+1) - (\mu^2)$
$(X_{n} - (\mu*n))^2 - \sigma^2 + \sigma^2n + \sigma^2 - (\mu^2)$
$(X_{n} - (\mu*n))^2 + \sigma^2n - (\mu^2)$
And I cant seem to get rid of the mu term.