Sequences & Series
AP — rationalizing surd sums
nta_pyq_2023_jan
Grade 11
Question:
Let $a_1, a_2, \ldots, a_n$ be in A.P. If $a_5 = 2a_7$ and $a_{11} = 18$, then $12\left(\dfrac{1}{\sqrt{a_{10}} + \sqrt{a_{11}}} + \dfrac{1}{\sqrt{a_{11}} + \sqrt{a_{12}}} + \ldots + \dfrac{1}{\sqrt{a_{17}} + \sqrt{a_{18}}}\right)$ is equal to ______.
Step-by-Step Solution
Key Concept: Find $a_1$ and $d$ from the given conditions. Rationalize each term: $\frac{1}{\sqrt{a_k}+\sqrt{a_{k+1}}} = \frac{\sqrt{a_{k+1}}-\sqrt{a_k}}{d}$ (telescoping).
$d = 9$, $a_1 = -72$... (recalculate: $a_5 = 2a_7 \Rightarrow a_1+4d = 2a_1+12d \Rightarrow a_1 = -8d$; $a_{11} = a_1+10d = 2d = 18 \Rightarrow d=9$). Sum telescopes to $\frac{\sqrt{a_{18}}-\sqrt{a_{10}}}{d}$. Answer $= 8$.
Correct Answer: 8