Sets, Relations & Functions
Range of a Relation / Geometric Series
nta_pyq_2025_apr
Grade 11
Question:
Let $S = \mathbb{N} \cup \{0\}$. Define a relation $R$ from $S$ to $\mathbb{R}$ by $R = \left\{(x,y) : \log_e y = x \log_e\left(\frac{2}{5}\right),\, x \in S,\, y \in \mathbb{R}\right\}$. Then, the sum of all elements in the range of $R$ is equal to:
$\frac{10}{9}$
$\frac{3}{2}$
$\frac{5}{2}$
$\frac{5}{3}$
Step-by-Step Solution
Key Concept: The range consists of $y = (2/5)^x$ for $x = 0, 1, 2, \ldots$ — a geometric series with ratio $2/5 < 1$.
Range $= \{(2/5)^0, (2/5)^1, (2/5)^2, \ldots\} = \{1, 2/5, 4/25, \ldots\}$. Sum $= \frac{1}{1-2/5} = \frac{5}{3}$.
Correct Answer: $\frac{5}{3}$