Consider a sequence of random variables $X_1, X_2, X_3, ..., X_n$, with $\mathbb{E}[X_i] = 0$ and $ \mathbb{E}[X_i^2] = \sigma^2 $, for all $ i = 1,2,...,n $. Let $\mathcal{F}_t := \sigma (X_1, X_2, ..., X_i)$ be the $\sigma$-algebra generated by $X_1, X_2, ..., X_i$.
Now I want to compute $\mathbb{E} [ (\sum_{i=1}^n X_i)^2 ]$ using the language of conditional expectations, and I find something awkward. Perhaps it's just I'm missing something trivial.
Here it goes. By definition of variance and the tower law of total expectation, we have \begin{align} & \mathbb{E} \left[ \left(\sum_{i=1}^n X_i \right)^2 \right] = \mathbb{E} \left[ \mathbb{E} \left[ \left(\sum_{i=1}^n X_i \right)^2 | \mathcal{F}_{n-1} \right] \right] = \mathbb{E} \left[ \mathbb{E} \left[ X_n^2 + \sum_{i=1}^{n-1} X_n X_i + \sum_{i,j<n} X_i X_j | \mathcal{F}_{n-1} \right] \right] \end{align} For $i < n$, $ \mathbb{E} [ X_n X_i | \mathcal{F}_{n-1} ] = 0$, since $X_i$ is effectively a constant given the information in $\mathcal{F}_{n-1}$.
Thus we have \begin{align} &\mathbb{E} \left[ \left(\sum_{i=1}^n X_i \right)^2 \right] = \mathbb{E} \left[ \mathbb{E} \left[ X_n^2 + \sum_{i,j<n} X_i X_j | \mathcal{F}_{n-1} \right] \right] = \sigma^2 + \mathbb{E} \left[ \mathbb{E} \left[ \left( \sum_{i=1}^{n-1} X_i \right)^2 | \mathcal{F}_{n-1} \right] \right] \\=& \sigma^2 + \mathbb{E} \left[ \left( \sum_{i=1}^{n-1} X_i \right)^2 \right]. \end{align}
Inductively, we get $ \mathbb{E} \left[ \left(\sum_{i=1}^n X_i \right)^2 \right] = n \sigma^2 $. The covariance terms all disappear! Why? Which step went wrong?