GATE 2024 ST – Question 12

Calculus · Sequences and Series · 1 mark · Multiple choice

Let a₁=√6 and a_(n+1)=√(6+a_n). I:the sequence increases. II:its limit is 2. Which hold?

Only I
Only II
Both I and II
Neither I nor II

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Show answer and explanation

Correct answer: (A) Only I

Explanation

Starting below 3, induction gives a_n<3. Also √(6+a_n)>a_n for 0<a_n<3 because a_n²−a_n−6=(a_n−3)(a_n+2)<0. Thus it increases to a finite L satisfying L²−L−6=0. Positivity gives L=3, not 2. **A**.

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