Recurring Theme, Part $n+1$

Algebra Level 5

If $a_1=10, a_2=6$ and $5a_{n+2}=6a_{n+1}-5a_n$ for $n>0$ , find the maximal value of $a_n$ , for all positive integers $n$ .

If you come to the conclusion that no such maximum exists, enter 666.

1 solution

A Former Brilliant Member
Mar 30, 2016

Using induction, it follows that $a_n=10\cos ((n-1)\alpha)$

where $\alpha \in \left(0,\frac{\pi}{2}\right)\quad and \quad \cos{\alpha}=\frac{3}{5}$

Therefore, $a_n\leq10\quad and \quad \text{equality holds when}\,n=1$

So, $\max_{n \in \mathbb{N}}\{a_n\}=a_1=\boxed{10}$

Or you simply use EXCEL!!!

Andreas Wendler - 5 years, 2 months ago

How do you plan to find the maximum from a infinite set of values using a computer program?

A Former Brilliant Member - 5 years, 2 months ago

I am quite able to realize this! -> Experience, trend etc.

Andreas Wendler - 5 years, 2 months ago

how did u prove that using induction .could u please elaborate .thanks

Zerocool 141 - 4 years, 7 months ago