4.1. Real numbers $a, b, c, d, e, f, g$ satisfy the equations

$$
\left\{\begin{array}{l}
a+4 b+9 c+16 d+25 e+36 f+49 g=1 \\
4 a+9 b+16 c+25 d+36 e+49 f+64 g=12 \\
9 a+16 b+25 c+36 d+49 e+64 f+81 f=123
\end{array}\right.
$$

Find $16 a+25 b+36 c+49 d+64 e+81 f+100 g$.
4.1. Answer: $3 \cdot 123-3 \cdot 12+1=334$.

Notice that $(n+2)^{2}-3(n+1)^{2}+3 n^{2}-(n-1)^{2}=0$, so $(16 a+25 b+36 c+49 d+64 e+81 f+100 g)-3(9 a+16 b+25 c+$ $36 d+49 e+64 f+81 f)+3(4 a+9 b+16 c+25 d+36 e+49 f+64 g)-(a+4 b+9 c+16 d+25 e+36 f+49 g)=0$.