Which is the smallest natural number n to which it applies a(n)>10?

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Nikodin Sedlarevic 2021-11-18

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编辑： John D'Errico 2021-11-18

I have recursive sequence a(i)= a(i-1) + a(i-2)^(-1), where is a(1) and a(2) equals 1. So I have to find smallest natural number n to which it applies a(n)>10.

This is my code so far:

a = zeros(1,15);

a(1) = 1;

a(2) = 1;

for i = 3:15

a(i)= a(i-1) + a(i-2)^(-1);

end

Any help?

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Stephen23 2021-11-18

@Nikodin Sedlarevic: that recursive sequence does not use b anywhere. What is b for?

Nikodin Sedlarevic 2021-11-18

Sorry, I will remove it

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Answer 1

David Hill 2021-11-18

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a(1)=1;a(2)=1;
for k=3:1000 %pick something to overshoot or use while loop
  a(k)=a(k-1)+1/a(k-2);
end
n=find(a>10,1);%48!

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Answer 2

John D'Errico 2021-11-18

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https://ww2.mathworks.cn/matlabcentral/answers/1590344-which-is-the-smallest-natural-number-n-to-which-it-applies-a-n-10#answer_835204

编辑：John D'Errico 2021-11-18

在 MATLAB Online 中打开

The obvious answer is brute force, thus...

a_i = 1;

a_iplus1 = 1;

plot(a_i,a_iplus1,'o')

hold on

iter = 2;

imax = 1e6;

while (a_iplus1 < 10) && (iter < imax)

iter = iter + 1;

[a_i,a_iplus1] = deal(a_iplus1,a_iplus1 + 1/a_i);

plot(a_i,a_iplus1,'o')

end

grid on

xlabel a_i

ylabel a_iplus1

a_iplus1

a_iplus1 = 10.0922

iter

iter = 48

So when iter = 48, a finally grows larger than 10.

Note that I wrote the code so that no arrays are grown. This is important, since the loop might have gone on forever. This is why I put an upper limit on the loop. The trick using deal to advance the terms is well, just a cute trick.

Does a general analytical solution to this nonlinear difference equation exist? Possibly. Such nonlinear difference equations tend to have "interesting" behaviour. As soon as you dive into the domain of the nonlinear, things can go straight to well, you know where. The plot however, suggests a simple asymptotic behavior, one that makes sense when you look at the expression. Questions now come up, like is there a limiting value? Or will a(i) grow forever, unbounded?