Sequences & Series
Sum of infinite series; logarithmic form
MJMT_Full_Test_11
Grade 12
Question:
If sum of the series $1+\dfrac{\sqrt5-\sqrt3}{2\sqrt5}+\dfrac{8-2\sqrt{15}}{30}+\dfrac{14\sqrt5-18\sqrt3}{60\sqrt5}+\cdots=2+\dfrac{a+\sqrt{15}}{b}\log_b\!\left(\dfrac{a}{c}\right)$; $a,b,c\in\mathbb{N}$, $\gcd(a,b,c)=1$, then $2(a^2+b^2+c^2)$ is
Step-by-Step Solution
Key Concept: Let $t=\frac{\sqrt5-\sqrt3}{\sqrt5}$. Rewrite $S$ as a standard logarithmic series: $\sum t^n/n(n+1)$ type, then sum using $-\ln(1-t)$ identity.
$2(a^2+b^2+c^2)=100$.
Correct Answer: 3