Hint for proving $\text{Var}\left(\frac{1}{n}\sum_{i=1}^{n}X_i\right)=ρσ^2+\frac{1-ρ}{n}σ^2$

Question

So this is part of a homework at university, which means, I obviously don't want a complete solution as answer, but only a hint as to what I do wrong.

We can assume $X_i$ to be one of $n$ identically distributed random variables; $\rho$ is the "positive pairwise correlation" between them. Also, $\text{Var}\left(X_i\right) = \sigma^2$. Now it is to prove that $\text{Var}\left(\frac{1}{n}\sum_{i=1}^{n}X_i\right)=ρσ^2+\frac{1-ρ}{n}σ^2$.

This is my apparently incorrect approach, which I want a hint for:

Answer 1

LHS of $(3)$ does not equal LHS of $(4)$.

By the summation you must discern the cases $i=j$ and $i\neq j$.

Answer 2

Hint: using $Var\left(\sum_{i=1}^{n} a_i X_i\right) = \sum_{i=1}^{n} a_i^2Var(X_i) + 2\sum_{1 \leq i < j \leq n} a_ia_j Cov(X_i, X_j)$ equation. Know more here.