Consider the recursive function $C_n = C_{n-1} + iC_{n-2}$, where $C_1 = 1, C_2 = 1.$ If $C_{10}$ is written in the form $a+bi,$ find $b$.

I solved this problem through brute force with a calculator. Is there an elegant solution method? I wasn't able to find any pattern in the terms, nor was I able to find a nice geometric solution.

  • $\begingroup$ It's a linear recurrence, we can easily find closed form of $C_n$. $\endgroup$ – Jakobian Oct 17 '18 at 19:56
  • $\begingroup$ Have you learned about recurrence relations? $\endgroup$ – saulspatz Oct 17 '18 at 19:59
  • $\begingroup$ I've seen them briefly before in an intro combinatorics class, but I was under the impression that it wouldn't work in this case because we have imaginary coefficients. $\endgroup$ – math783625 Oct 17 '18 at 20:00
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    $\begingroup$ Note; I'm not sure that the closed form solution helps all that much. The roots of the associated quadratic aren't very nice. Given that you only want $C_{10}$, I'd just do it recursively. $\endgroup$ – lulu Oct 17 '18 at 20:06
  • $\begingroup$ @math783625 The math works fine with complex numbers, but that doesn't help much in this case. I wrote my comment before solving the equation. I agree with others comments that the exact solution is too messy to be useful. $\endgroup$ – saulspatz Oct 17 '18 at 20:10

The comment of @lulu is most apposite. The equations are a mess, but once I caught my mistakes, the symbolic calculator worked like a charm. It’s perfectly standard recurrence-fu, and here it is:

You have a function $n\mapsto C_n$ that satisfies the stated recursion, and you make the educated guess that it’ll be a linear combination of exponentials, of which a basic one looks like $n\mapsto T^n$ for some value of $T$. The recurrence says $T^{n+2}=T^{n+1}+iT$, from which you immediately factor out $T^n$ to get the quadratic $T^2-T-i=0$. Roots of course are $$ t=\frac{1+\sqrt{1+4i}}2\,,\qquad t'=1-t_1\,, $$ and you must absolutely not try to go numerical at this stage.

You expect that $C_n=At^n+B(1-t)^n$, and by trying this with $C_0=1$ and $C_1=1$, you get $A=t/(2t-1)$ and $B=(t-1)/(2t-1)$. This gives the closed-form description $$ C_n=\frac{t^{n+1}}{2t-1}\>-\> \frac{(1-t)^{n+1}}{2t-1}\,. $$ Only now do you call in your symbolic calculator, and find that when $n=10$, you get, lo and behold, $C_{10}=-12-25i$.


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