A stock's correlation with its own index

An index holds $n$ stocks in equal weights. Each stock's daily return has the same volatility $\sigma$.

(a) Suppose the stocks are uncorrelated with each other. What is the correlation between one stock and the index? Give the value for $n = 25$ and describe it as an angle.

(b) Now let every pair of stocks have correlation $\rho = 0.25$, with $n = 100$. What is the correlation between one stock and the index, and what does it approach as $n$ grows?