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  • $s_{0} = s_{1} + 1\rightarrow(1)$
  • $2s_{i} = s_{i-1} + s_{i+1} +2\:\:\: \text{for}\:1\leq i \leq 8\rightarrow(2)$
  • $2s_{9} = s_{8} + 2\rightarrow(3)$

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 $(C).$

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$2s_9 = s_8 + 2$ $\rightarrow (0)$

$2s_9 = s_8 + 2 \Rightarrow s_9 = \frac{s_8 +2}{2}$

$2s_8 = s_7 + s_9 + 2 \Rightarrow s_9 = 2s_8 -s_7 -2$

So, $\frac{s_8 +2}{2} =  2s_8 -s_7 -2 $

$\Rightarrow 3s_8 = 2s_7 +6 $  $\rightarrow (1)$

Now,   $2s_7 = s_6 + s_8 + 2$   $\rightarrow (A)$

From equation $(1)$ and $(A),$ after eliminating $s_8$,

$\Rightarrow 4s_7 = 3s_6 +12 $  $\rightarrow (2)$

Now,   $2s_6 = s_5 + s_7 + 2$   $\rightarrow (B)$

From equation $(2)$ and $(B),$ after eliminating $s_7$,

$\Rightarrow 5s_6 = 4s_5 + 20 $  $\rightarrow (3)$

On following the pattern in equations $(0),(1),(2),(3),$ we get

$\Rightarrow 6s_5 = 5s_4 + 30 $

$\Rightarrow 7s_4 = 6s_3 + 42 $

$\Rightarrow 8s_3 = 7s_2 + 56 $

$\Rightarrow 9s_2 = 8s_1 + 72 $

Finally, $\Rightarrow 10s_1 = 9s_0 + 90 $

It is given that  $s_0 = s_1 + 1.$ So putting the value of $s_1= s_0 -1 $ in equation $10s_1 = 9s_0 + 90 $

$10(s_0 - 1) = 9s_0 +90 \Rightarrow s_0 =100$
Answer:

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