Explanation of Specific Sequence from Maths Homework

Question

As part of my homework, I have to find the next three terms in a sequence and explain my method. I am struggling to find the rule which links the terms:

1, 4, 13, 40, _, _, _

If anyone has any suggestions, it'd be very helpful - thanks in advance!

Answer 1

No wonder you're struggling! I'm sorry to say that this problem (at least as you've written it here) doesn't have an answer! Or, at least, it doesn't have a single, correct answer. You can make up your own, any way you'd like. The only criteria are, i think,

• There should be some rule governing how each element of the list appears
• The rule rule probably ought to give only one answer, so anyone reading that rule, anywhere and anywhen, as long as they understand it, should get the exact same sequence
• That rule shouldn't be broken by the start of the list (1, 4, 13, 40)
• The rule should be able to give at least three more terms (although it's probably meant to be possible to continue indefinitely)
• The next three terms you give should also follow this rule

Apart from that, you're free to pick whatever you want! If you manage to find a rule which gives the sequence $1, 4, 13, 40,-42,\pi, \sqrt 2$, then by all means, go for it.

That being said, there might be one specific rule that the problem authors intend you to find. I think that that's the one hinted at in the comments. That's also probably the easiest rule to follow, so you can go with that if you want.