Sets, Relations & Functions
Functional Symmetry / Sum of Series
nta_pyq_2025_apr
Grade 11
Question:
Let $f(x) = \dfrac{4^{x+2} + 16}{2 \cdot 2^{2x+1} + 2^{x+4} + 32}$. Then the value of $8\!\left(f\!\left(\frac{1}{15}\right) + f\!\left(\frac{2}{15}\right) + \cdots + f\!\left(\frac{59}{15}\right)\right)$ is equal to:
Step-by-Step Solution
Key Concept: Simplify $f(x)$ to find a symmetry $f(x)+f(4-x)=1/2$. Pair up terms in the sum.
$f(x) = \frac{2}{2^x+4}$. $f(x)+f(4-x)=\frac{1}{2}$. Terms pair as $f(k/15)+f((60-k)/15)=1/2$ for $k=1,\ldots,29$ (29 pairs) plus $f(30/15)=f(2)=\frac{2}{8}=\frac{1}{4}$. Sum $= 29\times\frac{1}{2}+\frac{1}{4}$. $8\left(\frac{29}{2}+\frac{1}{4}\right) = 8\times\frac{59}{4} = 118$.
Correct Answer: 118