# Another Number Sequence Problem

Find the next number in the sequence:

$$10,11,21,32,53,85,49,45,5,50,?$$

This sequence seems to me as possibly difficult to guess, so below is a hint.

Well, let me try...

(10 + 11) % 89 = 21 % 89 = 21
(11 + 21) % 89 = 32 % 89 = 32
(21 + 32) % 89 = 53 % 89 = 53
(32 + 53) % 89 = 85 % 89 = 85
(53 + 85) % 89 = 138 % 89 = 49
(85 + 49) % 89 = 134 % 89 = 45
(49 + 45) % 89 = 94 % 89 = 5
(45 + 5) % 89 = 50 % 89 = 50

So, the next one should be...

(5 + 50) % 89 = 55 % 89 = 55

• That is correct! I was thinking of $10^n\% 89$, but your pattern also works. – Jason Kim May 2 '19 at 4:25
• $10^n~\%~89 = 100 \times 10^{n-2}~\%~89 = 11 \times 10^{n-2}~\%~89 = (10^{n-1} + 10^{n-2})~\% 89$ – athin May 2 '19 at 4:31
• I noticed that when I thought about why your solution worked. If I had seen that pattern, I could have computed the terms so much easier... – Jason Kim May 2 '19 at 5:23