- $s_{0} = s_{1} + 1\rightarrow(1)$
- $2s_{i} = s_{i-1} + s_{i+1} +2\:\:\: \text{for}\:1\leq i \leq 8\rightarrow(2)$
- $2s_{9} = s_{8} + 2\rightarrow(3)$
Now,
- $2s_{8} = s_{7} + s_{9} + 2$
- $2s_{7} = s_{6} + s_{8} + 2$
- $2s_{6} = s_{5} + s_{7} + 2$
- $2s_{5} = s_{4} + s_{6} + 2$
- $2s_{4} = s_{3} + s_{5} + 2$
- $2s_{3} = s_{2} + s_{4} + 2$
- $2s_{2} = s_{1} + s_{3} + 2$
- $2s_{1} = s_{0} + s_{2} + 2$
Adding all the terms, and we get
$2(s_{8} + s_{7} + s_{6} + s_{5} + s_{4} + s_{3} + s_{2} + s_{1}) = s_{7} + s_{9} + 2+s_{6} + s_{8} + 2+s_{5} + s_{7} + 2+s_{4} + s_{6} + 2+ s_{3} + s_{5} + 2+s_{2} + s_{4} + 2+s_{1} + s_{3} + 2+s_{0} + s_{2} + 2$
$\implies s_{8} + s_{1} = s_{9} + s_{0} + 16$
Given that$:2s_{9} = s_{8} + 2\implies s_{8} = 2s_{9} - 2$
$\implies 2s_{9} - 2 + s_{0} - 1 = s_{9} + s_{0} + 16$
$\implies s_{9} = 19\rightarrow(4)$
Now, again
$2s_{1} = s_{0} + s_{2} + 2$
From the equation $(1),$ we get
$s_{1} = s_{0} - 1$
$2(s_{0} - 1) = s_{0} + s_{2} + 2$
$\implies 2s_{0}-2 = s_{0} + s_{2} + 2$
$\implies s_{0} = s_{2} + 4 $
$\implies s_{2} = s_{0} - 4(1+3)$
$2s_{2} = s_{1} + s_{3} + 2$
$2(s_{0} - 4) = s_{0} - 1 + s_{3} + 2$
$\implies 2s_{0} - 8 = s_{0} - 1 + s_{3} + 2$
$\implies s_{0} = s_{3} + 9$
$\implies s_{3} = s_{0} - 9(4+5)$
$2s_{3} = s_{2} + s_{4} + 2$
$2(s_{0} - 9) = s_{0} - 4 + s_{4} + 2$
$\implies 2s_{0} - 18 = s_{0} - 4 + s_{4} + 2$
$\implies s_{0} = s_{4} + 16$
$\implies s_{4} = s_{0} - 16(9+7)$
Similarly,
$\implies s_{5} = s_{0} - 25(16+9)$
$\implies s_{6} = s_{0} - 36(25+11)$
$\implies s_{7} = s_{0} - 49(36+13)$
$\implies s_{8} = s_{0} - 64(49 + 15)$
$\implies 2s_{9} - 2 = s_{0} - 64$
Put the value of $s_{9}$ from the equation $(4)$ and we get
$s_{0} = 2(19) - 2 + 64 = 36+64 = 100.$
So, the correct answer is $(C).$