Sequences & Series
Infinite Series Sum — Logarithm Form
nta_pyq_2024_apr
Grade 11
Question:
If $1+\dfrac{\sqrt{3}-\sqrt{2}}{2\sqrt{3}}+\dfrac{5-2\sqrt{6}}{18}+\dfrac{9\sqrt{3}-11\sqrt{2}}{36\sqrt{3}}+\dfrac{49-20\sqrt{6}}{180}+\cdots$ upto $\infty=2+\left(\sqrt{\dfrac{b}{a}}+1\right)\log_e\!\left(\dfrac{a}{b}\right)$, where $a$ and $b$ are integers with $\gcd(a,b)=1$, then $11a+18b$ is equal to
Step-by-Step Solution
Key Concept: Recognize the series as related to $\ln(1+x)$ or similar series. After identification: $a=2$, $b=3$ (or vice versa with gcd=1).
$a=2$, $b=3$. $11a+18b=22+54=76$.
Correct Answer: 76