(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

I don't remember the exact problems but I'll try to recall it as best as I can.

Given two positive real sequences a[itex]_{n}[/itex], b[itex]_{n}[/itex], with a[itex]_{1}[/itex] = b[itex]_{1}[/itex] = 1, and b[itex]_{n}[/itex] = b[itex]_{n-1}[/itex]a[itex]_{n}[/itex] - 2. Show that [itex]\sum^{\infty}_{n=2}[/itex] [itex]\frac{1}{a_{1}a_{2}\ldotsa_{n}}[/itex] converges and find what it converges to.

3. The attempt at a solution

To show that it converges, I tried to show that all a[itex]_{i}[/itex] from 2 to infinity have to be greater than 1. In other words, I want to show that

b[itex]_{n}[/itex] = b[itex]_{n-1}[/itex]a[itex]_{n}[/itex] - 2 > b[itex]_{n-1}[/itex] - 2.

So, I tried to show that by induction. First I had to find a[itex]_{2}[/itex], and found using the original sequence inequality that a[itex]_{2}[/itex] > 2 since all the b[itex]_{n}[/itex]'s are positive. Then I lost myself somewhere and just twiddled my thumbs for about 2 hours.

[a]1. This is going to be badly worded cause I'm using memory recall but: For an n x n matrix with integer values, find n for the matrix such that when you dot product a row vector to itself, you get an even number and when you multiply it to any other row vector in that matrix, you get an odd number.

3. I tried to find it by calling that initial matrix A and multiplying it to itself but I think I should have multiplied it to A[itex]^{t}[/itex] and got a matrix B with even numbers along the diagonal and odd numbers off the diagonal (to fit the two properties given), then tried to think of a way to find n using that matrix B. In other words, I had a staring contest with a blank piece of paper.

Anyone have any ideas/solutions, mostly interested in the how to tackle such problems and show a proper proof for them and why they sell hot dogs in packages of 10 and buns in packages of 8?

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Homework Help: Competition problems: 1. sequences/convergence 2. matrices

Can you offer guidance or do you also need help?

Draft saved
Draft deleted

**Physics Forums | Science Articles, Homework Help, Discussion**