China Mathematical Competition (Jiangxi)

The terms in the expression $f(x)$ form a geometric series with first term $1$ and common ratio $-x$. By the summation formula of geometric series, $$f(x)=\frac{(-x{)}^{21}-1}{-x-1}=\frac{x^{21}+1}{x+1}.$$ Set $x=y+4$, $g(y)=\frac{(y+4{)}^{21}+1}{y+5}$. Let $y=1$, we get $$a_0+a_1+\cdots +a_{20}=g(1)=\frac{5^{21}+1}{6}.$$