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Q. Assertion–Reason Type Question
Statement-1: \( \sqrt{5\sqrt{5\sqrt{5\cdots}}} = 5\sqrt{5} \)
Statement-2: \( \sqrt{x\sqrt{x\sqrt{x\cdots}}} = x,\ x>0 \)
✏️ Solution
Let \( y = \sqrt{5\sqrt{5\sqrt{5\cdots}}} \)
Then \( y = \sqrt{5y} \)
Squaring: \( y^2 = 5y \)
\( y(y-5) = 0 \Rightarrow y = 5 \) (since positive)
So LHS = 5, not \( 5\sqrt{5} \)
Statement-1 is FALSE
For Statement-2:
Let \( y = \sqrt{x\sqrt{x\cdots}} \Rightarrow y = \sqrt{xy} \)
\( y^2 = xy \Rightarrow y = x \)
So Statement-2 is TRUE
Correct Option: (d)
\( \boxed{\text{(d)}} \)