Sequence \(a_{n}\) where \(n = 1, 2, 3, \ldots\) is defined by \(a_{1} > 0\) and \(a_{n+1} = c(a_{n})^2 + a_{n}\) for \(n = 1,2,3,\ldots\) where \(c\) is constant, show that:

\(1.\) \( a_{n} \geq \sqrt{c^{n+1}n^n(a_{1})^{n+1}}\)

\(2.\) \(a_{1} + a_{2} + a_{3} + \ldots + a_{n} > n(na_{1} - \frac{1}{c})\) for \(n ∈ ℕ\)

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TopNewestNumber \(1\) isn't true for any \(c\), take \(n=1\), then \(1.\) reduces to

\(a_1\ge c a_1\)

, which is only true if \(c \le 1\). – A L · 3 years, 10 months ago

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– Colby Wolfe · 3 years, 10 months ago

And what's wrong with c being less than or equal to 1?Log in to reply

– Michael Tong · 3 years, 10 months ago

Well the wording of the problem would make one assume that it works for any constant...Log in to reply

– Colby Wolfe · 3 years, 10 months ago

Ah. I misread Alyosha's post as "there is no c such that it is true", but it was actually "it is not true for an arbitrary c". Thank you for the correction.Log in to reply

@Fariz , you must be sure of the validity before posting. – Jatin Yadav · 3 years, 10 months ago

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