UPSC NDA 2 2022 Mathematics Solved Paper

Concept:The functional equation $f(x+y)=f(x)f(y)$ with $f(1)=2$ defines an exponential function $f(n)=2^n$.The sum $\sum_{x=2}^{n} f(x)$ becomes a geometric series of powers of 2.Explanation:Given $f(x+y)=f(x)f(y)$ for all natural numbers and $f(1)=2$.Compute $f(2)=f(1+1)=f(1)f(1)=2\cdot2=4=2^2$.Compute $f(3)=f(2+1)=f(2)f(1)=4\cdot2=8=2^3$.By induction, $f(n)=2^n$ for all $n\in\mathbb{N}$.Thus $\sum_{x=2}^{n} f(x) = \sum_{x=2}^{n} 2^x = 2^2+2^3+\cdots+2^n$.This is a geometric progression with first term $a=4$, common ratio $r=2$, and number of terms $n-1$.Sum of GP: $S = a\frac{r^{n-1}-1}{r-1} = 4\frac{2^{n-1}-1}{2-1} = 4(2^{n-1}-1)$.Set equal to $2044$: $4(2^{n-1}-1)=2044$.Divide by 4: $2^{n-1}-1=511$.Add 1: $2^{n-1}=512=2^9$.Equate exponents: $n-1=9$, so $n=10$.Answer:$n=10$ (option C).