is there an answer to this? the solution can’t be too convoluted as it’s for a primary school kid.
it’s embarassing to be the go-to guy and not be able to give an answer right away. :smack:
359
2 * 2 + 1 = 5
5 * 3 + 2 = 17
17 *4 + 3 = 71
71 * 5 + 4 = 359
359 * 6 + 5 = 2159
ah thanks. how did you get the answer? did you just see the pattern or is there a method of approaching this besides trial and error?
Not that I would have got it anyway but I never had a chance because your example :
- 5, 17, 71, _____, 2159
I took it to be
2.5, 17, 71
Instead of
2, 5, 17, 71
oh, my bad. thanks for pointing it out.
Well, I assumed since it was for a 10-year-old it had to be pretty simple math. First I looked to see what I could do with 2 & 5 to make seventeen, but that didn’t for a pattern. So then I just looked at how to turn 2 into 5. And then 5 into 17. That formed a pattern.
That was a good get, Pochacco, I wonder if the pattern can be expressed any simpler?
I’m not buying it that this is standard math for a 10 year old!
It’s not the sort of question that appears on a test in class for a 10-year-old, but it is the sort of question that might appear on a book of mathematical teasers or even on some sort of I.Q.-like test. I would approach the question like this: The series is 2, 5, 17, 71, _____, 2159. Often the trick in series questiona is that the same number or some obvious series of numbers is added to each number to get the next number in the series. So look at additions:
5 = 2 + 3
17 = 5 + 12
71 = 17 + 54
So it doesn’t look like the same number is being added to each. The numbers being added are the series 3, 12, 54. There doesn’t appear to be any obvious connnection between the numbers in that series. So let’s try multiplication:
5 = 2 * 2 + something left over less than 2
17 = 3 * 5 + something left over less than 5
71 = 4 * 17 + something left over less than 17
So it’s like the numbers are multiplied by 2, 3, 4, etc. and then something else relatively small is added on. Let’s see what the numbers added on are:
5 = 2 * 2 + 1
17 = 3 * 5 + 2
71 = 4 * 17 + 3
This looks pretty good. Numbers from the series 2, 3, 4, etc. are multiplied and then numbers from the series 1, 2, 3, etc. are added. Let’s continue this:
5 = 2 * 2 + 1
17 = 3 * 5 + 2
71 = 4 * 17 + 3
359 = 5 * 71 + 4
2159 = 6 * 359 + 5
That’s exactly the number we want. So it looks like we have the answer the question wanted. This is the sort of question that would come up in a book of mathematical teasers for 10-year-olds.
I looked at the differences (3, 12, 54…) and figured there was something three-ish going on, so I added 1 to every term in the series to see if anything popped up, and it did. 
Thanks, Malacandra, for showing me that there’s an even better way of looking at the series. Look at it this way:
5 = 3 * 2 - 1
17 = 4 * 5 - 1
71 = 5 * 17 - 1
359 = 6 * 71 - 1
2159 = 7 * 359 - 1
So instead of noticing that each number is somewhat more than 2, 3, 4, 5, 6, etc. times the previous number, one can notice that the next number is 3, 4, 5, 6, 7, etc. times the previous number, minus 1.
I checked the On-Line Encyclopedia of Integer Sequences and the closest I saw was A082282.
So it’s not in the database. Perhaps the 10 year old can submit it and attain Internet immortality!
45 -1 = 19
517 -1 = 84
and so on
Yeah, thanks for that!
I was starting to think I’d gone crazy.
Yes - once you write down 3 6 18 72 — 2160 the increasing multiplier pretty much leaps off the page, giving you 360 to fill in the blank. Then for the original series you just subtract 1 from each term. 
ETA: I see you didn’t quite follow me above, but that’s how it works. 23 = 6, 36 = 18, 418 = 72, 572 = 360, 6*360 = 2160.
Admiral Crunch writes:
> 45 -1 = 19
> 517 -1 = 84
> and so on
You’re absolutely right. I was trying to simplify my formula from my first post and I blew it. This is what happens when you do the math in your head. What is really true if you look it as a subtraction is this:
5 = 3 * 2 - 1
17 = 4 * 5 - 3
71 = 5 * 17 - 14
359 = 6 * 71 - 67
2159 = 7 * 359 - 354
Now you can see the following:
1 = 2 - 1
3 = 5 - 2
14 = 17 - 3
67 = 71 - 4
354 = 359 - 5
That’s another way of generating the series, but it’s no easier than the previous method (and in fact it’s a little harder):
5 = 2 * 2 + 1
17 = 3 * 5 + 2
71 = 4 * 17 + 3
359 = 5 * 71 + 4
2159 = 6 * 359 + 5