JEE 2023 — Mathematics PYQ

$$x^n-y^n=(x-y)[x^{n-1}+x^{n-2}y+x^{n-3}{y}^2+\cdots +y^{n-1}]$$
$$(S_1)\to (2023{)}^{2022}-(1999{)}^{2022}$$
$$\to (2023)-(1999)=24\text{ is divisible by 8}$$
$$(S_2)\to (13)(1+12{)}^n-11n-13$$
$$=13[1+{}^n{C}_1(12)+{}^n{C}_2(12{)}^2+\dots ]-11n-13$$
$$\Rightarrow 145n+13\cdot {}^n{C}_2(12{)}^2+13\cdot {}^n{C}_3(12{)}^3+\dots$$