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

1. Dec 6, 2011

### muzak

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$_{n}$, b$_{n}$, with a$_{1}$ = b$_{1}$ = 1, and b$_{n}$ = b$_{n-1}$a$_{n}$ - 2. Show that $\sum^{\infty}_{n=2}$ $\frac{1}{a_{1}a_{2}\ldotsa_{n}}$ 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$_{i}$ from 2 to infinity have to be greater than 1. In other words, I want to show that

b$_{n}$ = b$_{n-1}$a$_{n}$ - 2 > b$_{n-1}$ - 2.

So, I tried to show that by induction. First I had to find a$_{2}$, and found using the original sequence inequality that a$_{2}$ > 2 since all the b$_{n}$'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$^{t}$ 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?