1. Originally Posted by vcs
Ugh.. yes. This is getting a bit complicated. Will get some sleep, clear my mind and think about it again tomorrow. Same for your reply in the programming thread as well, Ankit.
Originally Posted by silentstriker
I'm massively sleep deprived but: 0. Monkeys, Shakespeare, etc.

Right?
Originally Posted by weldone
In fact, for any x where 0<x<1, if you replace 3/4 by x and 1/4 by (1-x) in the original problem, then also p will come out as 1 from the equation I think. I'm also sleep-deprived, so not putting an effort to solve that generic one lol.
Originally Posted by weldone
I'll try that tommorrow maybe (assuming it doesn't need your continuous geometric algebra or stuff sounding similar to that ). Too sleepy now...
Originally Posted by nightprowler10
It's 2am so I might be missing something, but I get ln(2/5) evaluating the attached problem but the correct answer apparently is (1/3)ln(5/2). What the hell am I missing?

EDIT: oh god, ignore me.
worrying trend amongst our numb3r crunchers

2. slightly ashamed that i know what that means and even laughed

3. Window!!

4. Now here's some **** I understand

5. Originally Posted by nightprowler10
It's 2am so I might be missing something, but I get ln(2/5) evaluating the attached problem but the correct answer apparently is (1/3)ln(5/2). What the hell am I missing?

EDIT: oh god, ignore me.
aha we've all been there. The little things.

6. Prove that two consecutive numbers from Fibonacci Series are always relatively prime.

7. by contradiction and induction.

let f(n), f(n+1) be the nth fibonacci number for some integer n. suppose n=1. well... that's sort of trivial, i won't bother.

suppose true for some n=k ie. f(k) and f(k+1) are coprime.

suppose f(k+1) and f(k+2) are not coprime ie. there exists some non-trivial divisor d of both.
but f(k) = f(k+2) - f(k+1) so f(k) is also divisible by d. but this means that f(k), f(k+1) are not coprime - contradiction, qed, done.

8. Neat. Loved it.

9. Another question related to fibonacci numbers (I was asked this in an interview): What does the ratio of consecutive fibonacci number, f(k+1)/f(k) converge to as we go to as k->∞?

The part that was relatively easy was to find that ratio assuming convergence. But I don't known how to show there will be convergence in the first place. That will probably require university level calculus which I don't much remember (for every epsilon ε there exists a delta δ etc...)

EDIT: For limit of a sequence, you don't have δ but N.

10. Originally Posted by Spark
by contradiction and induction.

let f(n), f(n+1) be the nth fibonacci number for some integer n. suppose n=1. well... that's sort of trivial, i won't bother.

suppose true for some n=k ie. f(k) and f(k+1) are coprime.

suppose f(k+1) and f(k+2) are not coprime ie. there exists some non-trivial divisor d of both.
but f(k) = f(k+2) - f(k+1) so f(k) is also divisible by d. but this means that f(k), f(k+1) are not coprime - contradiction, qed, done.
Exactly.

Simple, isn't it? I've asked that question to 3 people while interviewing them yesterday. None of them got it right without any hint. (One guy cracked it after I suggested him to approach it through method of induction.)

11. Originally Posted by 8ankitj
Another question related to fibonacci numbers (I was asked this in an interview): What does the ratio of consecutive fibonacci number, f(k+1)/f(k) converge to as we go to as k->∞?

The part that was relatively easy was to find that ratio assuming convergence. But I don't known how to show there will be convergence in the first place. That will probably require university level calculus which I don't much remember (for every epsilon ε there exists a delta δ etc...)

EDIT: For limit of a sequence, you don't have δ but N.
It's very easy after assuming convergence.

f(k+1)/f(k) = {f(k)+f(k-1)}/f(k) = 1 + f(k-1)/f(k)
So, if the answer is x then x ~ (1 + 1/x)
=> x^2-x-1 = 0

Solving this quadratic and ignoring the value that is less than 1, we get
x = sqrt(5)/2 +0.5

12. Originally Posted by weldone
Exactly.

Simple, isn't it? I've asked that question to 3 people while interviewing them yesterday. None of them got it right without any hint. (One guy cracked it after I suggested him to approach it through method of induction.)
Good enough performance considering interview pressure, isn't it? I try to keep interview questions somewhat stress free and non-trick kind. Here's a question I have been asking for data analyst kind of roles:

Probability of observing an accident in one hour period on an accident prone bridge is 10%. What is the probability of observing an accident in two hour period?

Don't mind if the candidate answers it after some hints. Is that too lenient?

13. Originally Posted by 8ankitj
probability of observing an accident in one hour period on an accident prone bridge is 10%. What is the probability of observing an accident in two hour period?
0.19?

14. yeah that's just 1 - 0.9^2 innit?

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