Recent content by Arew

Can't think of anything else :) Thanks.

I was worried we proved w_k for every other integer k. I see why that's wrong. Thanks.

Thanks for the comments. Do we have to handle $$w_{k+1}$$ at all?

Homework Statement Prove by induction $$w_k = w_{k−2} + k$$, for all integers $$k \ge 3, w_1 = 1,w_2 = 2$$ has an explicit formula $$ w_n =\begin{cases} \frac{(n+1)^2}{4}, & \text{if $n$ is odd} \\ \frac n2(\frac n2 + 1), & \text{if $n$ is even} \end{cases}$$ Homework Equations The Attempt...

I was unsure. Usually when we find an explicit formula for a recurrence, we get rid of indices in the recurrence relation so that we do at most one substitution into the formula. But if you say I am done - I am happy to take a break :) As for the proof, I think induction should take care of it...

Oh, I never knew you had to put the dollar signs in pairs. Thanks.$$2^{F_k} = 2^{F_{k-1} + F_{k-2}}$$ then $$2^{F_n} = 2^ {\frac{\frac {1 + \sqrt 5}{2} - \frac {1 + \sqrt 5}{2}}{\sqrt 5}}$$ which implies $$\log_22^{F_n} = \log_22^ {\frac{\frac {1 + \sqrt 5}{2} - \frac {1 + \sqrt 5}{2}}{\sqrt...

Homework Statement Use iteration to guess an explicit formula for u_k = u_{k−2} * u_{k−1}, for all integers k ≥ 2, u_0 = u_1 = 2 and prove it . Homework Equations Hint: Express the answer using the Fibonacci sequence. The Attempt at a Solution u_k = u_{k−2} * u_{k−1} and u_0 = u_1 = 2, so...