Sequences and series (6)

Algebra Level 4

A sequence of positive terms A 1 , A 2 , A 3 , ⋯ , A n A_{1}, A_{2}, A_{3}, \cdots, A_{n} satisfies the relation A n + 1 = 3 ( 1 + A n ) ( 3 + A n ) A_{n+1} = \dfrac{3(1+A_{n})}{(3+A_{n})} . Find the least integral value of A 1 A_{1} for which the above sequence is decreasing.


The answer is 2.

Rahil Sehgal by Rahil Sehgal , 20, India

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1 solution

For a decreasing sequence ,

A 1 > A 2 A 1 > 3 + 3 A 1 3 + A 1 A 1 2 > 3 A ! > 3 \displaystyle \begin{aligned} &A_1\gt A_2 \\ &A_1 \gt \dfrac{3+3A_1}{3+A_1} \\ &A_1^2\gt 3 \\ &A_! \gt \sqrt{3}\end{aligned}

The least integral value corresponds to 2 \boxed{2}

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