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Confused with sequence

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 $$n ∈ ℕ$$

Note by Fariz Azmi Pratama
4 years, 5 months ago

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Number $$1$$ isn't true for any $$c$$, take $$n=1$$, then $$1.$$ reduces to

$$a_1\ge c a_1$$

, which is only true if $$c \le 1$$.

- 4 years, 4 months ago

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

- 4 years, 4 months ago

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

- 4 years, 4 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.

- 4 years, 4 months ago

True that!

@Fariz , you must be sure of the validity before posting.

- 4 years, 4 months ago