Question

Let $a_1, a_2, \ldots, a_n$ be in AP If $a_5=2 a_7$ and $a_{11}=18$, then $12\left(\frac{1}{\sqrt{a_{10}}+\sqrt{a_{11}}}+\frac{1}{\sqrt{a_{11}}+\sqrt{a_{12}}}+\ldots+\frac{1}{\sqrt{a_{17}}+\sqrt{a_{18}}}\right)$ is equal to

Hint:

In problems involving arithmetic progressions and sums of series, make sure to use the general term formula and simplify terms systematically to calculate the total sum efficiently.

Answer 1

Correct Answer: 8

The correct answer is 8.
$2a_5=a_5(\text{given})$
$2\left(a_1+6d\right)=a_1+4d$
$a_1+8d=0$....(1)
$a_1+10d=18$...(2)
$\text{By}(1)\text{and (2) we get}{a}_1=-72,d=9$
$a_{18}=a_1+17d=-72+153=81$
$a_{10}=a_1+9d=9$
$12\left(\frac{\sqrt{a_{11}}-\sqrt{a_{10}}}{d}+\frac{\sqrt{a_{12}}-\sqrt{a_{11}}}{d}+\dots \dots \frac{\sqrt{a_{18}}-\sqrt{a_{17}}}{d}\right)$
$12\left(\frac{\sqrt{a_{18}}-\sqrt{a_{10}}}{d}\right)=\frac{12(9-3)}{9}=\frac{12\times 6}{6}=8$

Answer 2

Given Equation:

\[ 2a_7 = a_5 \quad \text{(given)} \]

Step 1: Forming Equations

From the arithmetic sequence formula:

\[ 2(a_1 + 6d) = a_1 + 4d \]

Expanding and simplifying:

\[ a_1 + 8d = 0 \quad \dots (1) \] \[ a_1 + 10d = 18 \quad \dots (2) \]

Step 2: Solving for \( a_1 \) and \( d \)

Solving equations (1) and (2):

\[ a_1 = -72, \quad d = 9. \]

Step 3: Computing Terms

Finding \( a_{18} \):

\[ a_{18} = a_1 + 17d = -72 + 153 = 81. \]

Finding \( a_{10} \):

\[ a_{10} = a_1 + 9d = 9. \]

Step 4: Evaluating the Expression

We evaluate the given sum:

\[ 12 \left( \frac{\sqrt{a_{18}} - \sqrt{a_{10}}}{d} + \frac{\sqrt{a_{12}} - \sqrt{a_{11}}}{d} + \dots + \frac{\sqrt{a_{18}} - \sqrt{a_{17}}}{d} \right) \]

Simplifying the numerator:

\[ 12 \left( \frac{\sqrt{a_{18}} - \sqrt{a_{10}}}{d} \right) = \frac{12 (9 - 3)}{9} = \frac{12 \times 6}{6} = 8. \]

Final Answer:

\[ \boxed{8} \]