Jason Yuan's Math Blogs

Hi all! As many of you know, the AMC’s were in November and it was quite the experience. I want to share with you one of the problems on the test that demonstrates one of my favorite methods of solving sequence problems!

Here is the problem: The Fibonacci numbers are defined by F₁ = 1, F₂ = 1, and F_n = F_n-1 + F_n-2  for n > 2. What is F₂ / F₁ + F₄ / F₂  + F₆ / F₃ + … + F₂₀ / F₁₀?

(A) 318 (B) 319 (C) 320  (D) 321 (E) 322

It was placed number 23 on the AMC10 and number 18 on the AMC12. Now for the big reveal of what that technique is! Drumroll, please!

Writing out the first few terms and finding a pattern. That’s it. 

On the AMCs, there is no need to prove anything via induction or contradiction or anything like that: there’s no time. When you see a pattern, you can just assume it holds true. When I saw this problem, I wanted to skip it because solving it for real might be challenging and take too much time, but then I realized I didn’t have to solve it for real, lol. I wrote the first 4 terms and found the pattern. 

The first few terms of the Fibonacci sequence are 1, 1, 2, 3, 5, 8, 13, 21. 

F₂ / F₁ + F₄ / F₂  + F₆ / F₃ + F₈ / F₄ would then be 1 + 3 + 4 + 7. Maybe you see it, maybe you don’t. Writing the next 2 terms might help a bit:

1 + 3 + 4 + 7 + 11 + 18. It’s another Fibonacci-adjacent sequence! Each term is the sum of the previous two terms. Assuming this holds true, the 10 terms would be 

1 + 3 + 4 + 7 + 11 + 18 + 29 + 47 + 76 + 123. If you add them all up, you get 319 or B. 

There’s no mathematical lesson here or a clever trick, but I think if you’re reading this blog and are going to take a competition in the future, I would advise you to at least try all of the sequence problems you can, especially on the AMCs. Just writing out the first few terms and finding a pattern is something you can do very fast, and it will often get you the right answer. On the AMCs, I believe these problems are placed higher than they should be because it takes much longer to prove that the sequence will follow the pattern, but if you’re just looking to score high, there’s no need to do that. 

Goodbye, and happy bashing!