Question 30
a. Determine $a,b,c$ such that
$$(n+2)^2=a(n+1)^2+bn^2+c(n-1)^2$$
holds for every number $n$.
b. Suppose $x_1,x_2,\ldots,x_7$ satisfy
$$\begin{cases}x_1+4x_2+9x_3+16x_4+25x_5+36x_6+49x_7=1,\\4x_1+9x_2+16x_3+25x_4+36x_5+49x_6+64x_7=12,\\9x_1+16x_2+25x_3+36x_4+49x_5+64x_6+81x_7=123.\end{cases}$$
Find $16x_1+25x_2+36x_3+49x_4+64x_5+81x_6+100x_7$.
$$(n+2)^2=a(n+1)^2+bn^2+c(n-1)^2$$
holds for every number $n$.
b. Suppose $x_1,x_2,\ldots,x_7$ satisfy
$$\begin{cases}x_1+4x_2+9x_3+16x_4+25x_5+36x_6+49x_7=1,\\4x_1+9x_2+16x_3+25x_4+36x_5+49x_6+64x_7=12,\\9x_1+16x_2+25x_3+36x_4+49x_5+64x_6+81x_7=123.\end{cases}$$
Find $16x_1+25x_2+36x_3+49x_4+64x_5+81x_6+100x_7$.