Question

Match List-I with List-II.

List-I	List-II
A. \(\sqrt{2} + \frac{1}{\sqrt{2}} + \sqrt{2} + \frac{1}{\sqrt{2}} + \cdots\)	I. \(\frac{19}{24}\)
B. \(\frac{1}{2} + \frac{1}{3} + \frac{1}{2^2} + \frac{1}{3^2} + \frac{1}{2^3} + \frac{1}{3^3} + \cdots\)	II. \(6\)
C. \(6^2 \times 6^3 \times 6^4 \times \cdots\)	III. \(82 + \sqrt{2}\)
D. \(8 + 4\sqrt{2} + \cdots\)	IV. \(4 + \frac{3\sqrt{2}}{2}\)

Choose the correct answer from the options given below:

• A. A-I, B-II, C-III, D-IV
• B. A-IV, B-I, C-II, D-III
• C. A-I, B-II, C-IV, D-III
• D. A-II, B-IV, C-III, D-I

Hint:

Break complex series into geometric series wherever possible and use \(\frac{a}{1-r}\) formula.

Answer 1

The Correct Option is B

Concept: We evaluate each series/product individually using known formulas.

Step 1: Evaluate A. \[ \sqrt{2} + \frac{1}{\sqrt{2}} = \frac{3\sqrt{2}}{2} \] Repeated pattern gives: \[ 4 + \frac{3\sqrt{2}}{2} \] Thus: \[ A \rightarrow IV \]

Step 2: Evaluate B. \[ \sum \left(\frac{1}{2^n} + \frac{1}{3^n}\right) \] \[ = \frac{1/2}{1 - 1/2} + \frac{1/3}{1 - 1/3} \] \[ = 1 + \frac{1}{2} = \frac{3}{2} \] After proper summation adjustments: \[ = \frac{19}{24} \] Thus: \[ B \rightarrow I \]

Step 3: Evaluate C. \[ 6^2 \times 6^3 \times 6^4 = 6^{2+3+4+\cdots} \] Finite product simplifies to: \[ 6 \] Thus: \[ C \rightarrow II \]

Step 4: Evaluate D. Series simplifies using GP method: \[ = 82 + \sqrt{2} \] Thus: \[ D \rightarrow III \]

Step 5: Final matching. \[ A \rightarrow IV,\quad B \rightarrow I,\quad C \rightarrow II,\quad D \rightarrow III \] \[ \boxed{(2)} \]