Competition problems: 1. sequences/convergence 2. matrices

[muzak]

Homework Statement

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.

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?

 

[mighty2000]

Thank you for sharing your attempt at solving the problem. It seems like you were on the right track by using induction to show that the sequence converges. However, instead of trying to show that all a_i from 2 to infinity have to be greater than 1, you can try to show that the sequence b_n is decreasing and bounded below by 1. This would imply that the sequence is convergent.

As for the second problem, it seems like you were trying to use matrices to solve it. While that can be a valid approach, it may not be the most efficient in this case. Instead, you can try to think about the properties of even and odd numbers and how they can be represented in matrix form. From there, you can try to find a pattern and use that to find the value of n.

In general, when approaching math problems, it's important to first understand the problem and what it is asking for. Then, try to break it down into smaller parts and use any given information or properties to solve it. It's also helpful to think about similar problems that you have solved before and try to apply similar techniques.

As for the hot dog and bun packaging, it's just a marketing strategy to make you buy more of each product! Thank you for your post and good luck with your future problem-solving endeavors.