Hello r-soy
(1) $\displaystyle d = 3$. Correct.
$\displaystyle S_{51} = \frac{51}{2}\Big(2(3)+50(\color{red}3\color{black} )\Big) = 3978$
(2) The algebra in the induction proof is not complete. You need to say:$\displaystyle \tfrac13(4k^3-k)+(2(k+1)-1)^2 = \tfrac13(4k^3-k)+(2k+1)^2$$\displaystyle =\tfrac13(4k^3-k+3[4k^2+4k+1])$
$\displaystyle =\tfrac13(4k^3+12k^2+11k+3)$
$\displaystyle =\tfrac13(4k^3+12k^2+12k+4-[k+1])$
$\displaystyle =\tfrac13(4[k+1]^3-[k+1])$
(3) Again, your working is not complete:$\displaystyle \left(\frac{y}{x}\right)^{k+1}=\left(\frac{y}{x}\r ight)^{k}\cdot\frac{y}{x}$$\displaystyle =\frac{y^k}{x^k}\cdot\frac yx$
$\displaystyle =\frac{y^{k+1}}{x^{k+1}}$
Grandad