First, we know $s_n(x)=\frac{a_0}{2}+\sum_{k=1}^n(a_k\cdot \cos(kx)+b_k\cdot \sin(kx))$, $\{s_n\}$ is the sequence of partial sum of the fourier series generated by $f$, and $$\sigma_n(x)=\frac{s_0(x)+s_1(x)+\cdots+s_{n-1}(x)}{n}, \space (n=1,2,\ldots)$$ Want to show: $\exists$ coefficients $c_k, d_k \in C$ such that $$\sigma_n(x)=\frac{c_0}{2}+\sum_{k=1}^{n-1}(c_k\cdot \cos(kx)+d_k\cdot \sin(kx))$$and find $c_k, d_k$ in terms of the Fourier coefficients, $a_k, b_k$ of $f$.
My attempt: I listed $\sigma_1,\ldots$ and found $\sigma_n(x)=\frac{a_0}{2}+\sum_{k=1}^{n-1}(\frac{n-k}{n}a_kcos(kx)+\frac{n-k}{n}b_ksin(kx))$, that is $c_0=a_0, c_k=\frac{n-k}{n}a_k, d_k=\frac{n-k}{n}b_k$. Then, I plan to prove this by induction. Not sure if I am on the right track tho...