Continuous Probability and Central Limit Theorem -- Approximating sum of coin tosses distribution I ran into a problem I know I have to use central limit theorem and continuous probability for, but I'm not sure how to really apply those. Would appreciate some guidance :))

Assume there are 1 through k people with coins. Person i's coin has
sides i and -i. They toss their coins n times and sum the outputs into
a number $X_i$. For a large n, what is the distribution of $(X_1 + ... +
> X_k)/\sqrt{n}$ approximately equal?


Let's call the result of person $i$'s $j$th toss $Y_{i,j}$ which has mean $0$ and variance $i^2$.
Now let $Z_j = Y_{1,j}+\cdots + Y_{k,j}$, which will have mean $0$ and variance $\frac16 k(k+1)(2k+1)$.
Applying the central limit theorem to $Z_j$, we get that $\frac1{\sqrt{n}}(Z_1+ \cdots + Z_{n})$ converges in distribution to $\mathcal N\left(0, \frac16 k(k+1)(2k+1)\right)$.
Since  $Z_1+ \cdots + Z_{n} = X_1+\cdots +X_k$, you get that $\frac{X_1+\cdots +X_k}{\sqrt{n}}$ also converges in distribution to $\mathcal N\left(0, \frac16 k(k+1)(2k+1)\right)$.
