If 12 + 22 + 32 + . . . . . + p2 = [ p ( p + 1 ) ( 2 p + 1 )6 ] , then 12 + 32 + 52 + . . . . . + 172 is = ?

Answer: Option A

1² + 3² + 5² + . . . . . + 17²
= (1² + 2² + 3² + 4² + . . . . . + 17²) - (2² + 4² + . . . . . + 16²)
= (1² + 2² + 3² + 4² + . . . . . + 17²) - 2²(1² + 2² + . . . . . + 8²)
Using given formula
$$\begin{array}{rl}& \left[\frac{p\left(p+1\right)\left(2p+1\right)}{6}\right]\\ & =\frac{17\times 18\times 35}{6}-4\left[\frac{8\times 9\times 17}{6}\right]\\ & =1785-4\times 204\\ & =1785-816\\ & =969\end{array}$$