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I want to know the next number in the sequence below:

1935, 1940, 1948, 1962, 1985, 2020, …

Consecutive differences are 5, 8, 14, 23, 35

They don't look like Fib-seq. Also consecutive differences of the consecutive differences increase rapidly!

This is from math olympiad for middle school students. There are 4 options : 2070, 2055, 2060, 2067.

  • Any suggestions? Ask questions that can actually be answered. Questions of the form "what is the next number in the sequence" are entirely subjective. Any number can be the next and there is nothing that should suggest otherwise. It may as well be that the next number is $42$ and you can't prove me wrong. – JMoravitz Jan 16 '22 at 18:52
  • @JMoravitz I edited my post –  Jan 16 '22 at 18:58
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    Notice that there is a pattern in the second differences, $3,6,9,12,\cdots$. Does that correspond to one of the possible answers? – John Wayland Bales Jan 16 '22 at 19:01
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    I put in in OEIS and it said: Your sequence appears to be: $+ \frac12 x^3 − \frac32 x^2 + 6 x + 1930$ – GEdgar Jan 16 '22 at 19:49

1 Answers1

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If you find the consecutive difference of the consecutive differences you can see there is a pattern.

$1935, 1940, 1948, 1962, 1985, 2020$

consecutive differences are

$5,8,14,23,35$

consecutive differences of these consecutive differences

$3,6,9,12$

consecutive differences of those consecutive differences

$3,3,3$

assuming these 3s don't change the next number in the sequence is 2070.

Interestingly, if you subtract 1/2 n^3, you will cancel out these common differences and will be left with a quadratic sequence. By performing a similar process with the cube numbers, can you work out why this is the case?

Finlay Hutchinson
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