First of all you need the standard result of conditional distribution:

http://en.wikipedia.org/wiki/Multivariate_normal_distribution#Conditional_distributions
Intuition: It will be easy to see that

[math] E[X_n|X_2 = x_2] = E[X_n] + \frac {Cov[X_n, X_2]} {Var[X_2]} (x_2 - E[X_2]) [/math]

which is a standard result seen in regression.

Since this results holds [math] \forall x_2 \in \mathbb{R} [/math], and thus

[math] E[X_n|X_2] = E[X_n] + \frac {Cov[X_n, X_2]} {Var[X_2]} (X_2 - E[X_2])

[/math]

and it will be tempting to put the conditional [math] X_1 [/math] inside to reach the conclusion. To verify this we can do the following calculation.

To shorten the notation, first we write

[math] \begin{bmatrix} X_1 \\ X_2 \\ X_n \end{bmatrix} \sim \mathcal{N}

\left(\begin{bmatrix} \mu_1 \\ \mu_2 \\ \mu_n \end{bmatrix},

\begin{bmatrix} \sigma_1^2 & \sigma_{12} & \sigma_{1n} \\

\sigma_{12} & \sigma_2^2 & \sigma_{2n} \\

\sigma_{1n} & \sigma_{2n} & \sigma_n^2 \end{bmatrix} \right)[/math]

Then following the formula,

[math] E[X_n|X_1 = x_1, X_2 = x_2] [/math]

[math] = \mu_n + \begin{bmatrix} \sigma_{1n} & \sigma_{2n} \end{bmatrix}

\begin{bmatrix} \sigma_1^2 & \sigma_{12} \\ \sigma_{12} & \sigma_2^2 \end{bmatrix}^{-1}

\begin{bmatrix} x_1 - \mu_1 \\ x_2 - \mu_2 \end{bmatrix} [/math]

[math] = \mu_n + \begin{bmatrix} \sigma_{1n} & \sigma_{2n} \end{bmatrix}

\frac {1} {\sigma_1^2\sigma_2^2 - \sigma_{12}^2}\begin{bmatrix} \sigma_2^2 & -\sigma_{12} \\ -\sigma_{12} & \sigma_1^2 \end{bmatrix}\begin{bmatrix} x_1 - \mu_1 \\ x_2 - \mu_2 \end{bmatrix} [/math]

[math] = \mu_n + \frac {(\sigma_{1n}\sigma_2^2 - \sigma_{2n}\sigma_{12})(x_1 - \mu_1) + (\sigma_{2n}\sigma_1^2 - \sigma_{1n}\sigma_{12})(x_2 - \mu_2)} {\sigma_1^2\sigma_2^2 - \sigma_{12}^2} [/math]

On the other hand, the covariance matrix of [math] X_2, X_n|X_1 [/math] is

[math] \begin{bmatrix} \sigma_2^2 & \sigma_{2n} \\ \sigma_{2n} & \sigma_n^2 \end{bmatrix} -

\begin{bmatrix} \sigma_{12} \\ \sigma_{1n} \end{bmatrix}

\begin{bmatrix} \sigma_1^2 \end{bmatrix}^{-1}

\begin{bmatrix} \sigma_{12} & \sigma_{1n} \end{bmatrix} [/math]

[math] = \begin{bmatrix}

\displaystyle \sigma_2^2 - \frac {\sigma_{12}^2} {\sigma_1^2} &

\displaystyle \sigma_{2n} - \frac {\sigma_{12}\sigma_{1n}} {\sigma_1^2} \\

\displaystyle \sigma_{2n} - \frac {\sigma_{1n}\sigma_{12}} {\sigma_1^2} &

\displaystyle \sigma_n^2 - \frac {\sigma_{1n}^2}{\sigma_1^2} \end{bmatrix} [/math]

which means that

[math] Cov[X_2, X_n|X_1] = \sigma_{2n} - \frac {\sigma_{12}\sigma_{1n}} {\sigma_1^2} [/math]

Now consider

[math] \frac {Cov[X_n - \mu_n, X_2|X_1]} {Var[X_2|X_1]} \times (X_2 - E[X_2|X_1]) [/math]

[math] = \frac {\displaystyle \sigma_{2n} - \frac {\sigma_{12}\sigma_{1n}} {\sigma_1^2}}

{\displaystyle \sigma_2^2 - \frac {\sigma_{12}^2} {\sigma_1^2}}

\times \left\{X_2 - \left[\mu_2 + \frac {\sigma_{12}} {\sigma_1^2}(X_1 - \mu_1) \right]\right\} [/math]

[math] = \frac {\sigma_{2n}\sigma_1^2 - \sigma_{12}\sigma_{1n}} {\sigma_1^2\sigma_2^2 - \sigma_{12}^2}

\times \left[(X_2 - \mu_2) - \frac {\sigma_{12}} {\sigma_1^2}(X_1 - \mu_1) \right][/math]

[math] = \frac {\displaystyle \left(\frac {\sigma_{12}^2\sigma_{1n}} {\sigma_1^2} - \sigma_{2n}\sigma_{12}\right)(X_1 - \mu_1) + (\sigma_{2n}\sigma_1^2 - \sigma_{12}\sigma_{1n})(X_2 - \mu_2)} {\sigma_1^2\sigma_2^2 - \sigma_{12}^2} [/math]

Lastly,

[math] E[X_n|X_1] = \mu_n + \frac {\sigma_{1n}} {\sigma_1^2} (X_1 - \mu_1) [/math]

Combining together,

[math] E[X_n|X_1] + \frac {Cov[X_n - \mu_n, X_2|X_1]} {Var[X_2|X_1]} \times (X_2 - E[X_2|X_1]) [/math]

[math] = \mu_n + \frac {(\sigma_{1n}\sigma_2^2 - \sigma_{2n}\sigma_{12})(X_1 - \mu_1) + (\sigma_{2n}\sigma_1^2 - \sigma_{12}\sigma_{1n})(X_2 - \mu_2)} {\sigma_1^2\sigma_2^2 - \sigma_{12}^2} [/math]

which is the same as the expression calculated above.