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.
The sequence is about remainders.
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1 Answer
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3
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
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2$\begingroup$ That is correct! I was thinking of $10^n\% 89$, but your pattern also works. $\endgroup$ Commented May 2, 2019 at 4:25
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1$\begingroup$ $10^n~\%~89 = 100 \times 10^{n-2}~\%~89 = 11 \times 10^{n-2}~\%~89 = (10^{n-1} + 10^{n-2})~\% 89$ $\endgroup$– athinCommented May 2, 2019 at 4:31
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$\begingroup$ 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... $\endgroup$ Commented May 2, 2019 at 5:23