Sets, Relations & Functions
Exponential and Constant Functions — Summation
nta_pyq_2026_jan
Grade 11
Question:
Let $f$ and $g$ be functions satisfying $f(x+y)=f(x)f(y)$, $f(1)=7$ and $g(x+y)=g(xy)$, $g(1)=1$, for all $x,y\in\mathbb{N}$. If $\displaystyle\sum_{x=1}^n\left(\dfrac{f(x)}{g(x)}\right)=19607$, then $n$ is equal to:
Step-by-Step Solution
Key Concept: $f(x+y)=f(x)f(y)$ with $f(1)=7$: $f(x)=7^x$. $g(x+y)=g(xy)$ with $g(1)=1$: put $y=1$ to get $g(x+1)=g(x)$, so $g(x)=1$ for all $x\in\mathbb{N}$.
$n=5$.
Correct Answer: 4