ASMO Math 2022 — Grade 8 — School Round Question 24

Let $P(x)=a_0+a_1x+a_2x^2+\ldots+a_nx^n$ be a polynomial in x where the coefficients $a_0,a_1,\ldots,a_n$ are non-negative integers. If $P(1)=25$ and $P(27)=1771769$, determine $a_0+2a_1+3a_2+\ldots+(n+1)a_n$.