JEE 2023 — Mathematics PYQ

Solution

Let $f(x)=5\cdot k^x$

$5[5\cdot k^{x+y}]=[5\cdot k^x]\cdot [5\cdot k^y]⟹25\cdot k^{x+y}=25\cdot k^{x+y}$ (Verified)

$f(3)=5\cdot k^3=320⟹k^3=64⟹k=4$

$f(n)=5\cdot 4^n$

${\sum }_{n=0}^{5}5\cdot 4^n=5[4^0+4^1+4^2+4^3+4^4+4^5]$

$S=5\left[\frac{1(4^6-1)}{4-1}\right]=\frac{5}{3}(4096-1)=\frac{5\times 4095}{3}=5\times 1365=6825$

Sahi Option: (3)