## Friday, March 27, 2009

### Friday Fun: What's the next number in the sequence?

Can you figure out the next few terms in the following sequence?
Q: 1, 3, 7, 12, 18, 26, 35, 45, 56, 69...
I'll post the answer some time next week.

1. Still no comments?

This one isn't so hard. The twentieth term has 20 as a factor, along with a prime.

2. William has the correct answer to an elegant puzzle.

There is another series which, because of its clumsiness is obviously not the desired answer. Neverless, it is an answer to the question. Terms 11 thru 13 are 146,672,and 3083. (That is, if I didn't make an error.)

3. Now for extra credit:

Why didn't the twenty-first term get a comment?

4. The sum of the 21st term's factors coincidentally correspond to another number.

5. I believe the 11, 12, 13 numbers are : 83, 98, 114

6. Natasha is correct.

7. Are you reading between the lines?

8. Blaine’s reply to Natasha tells me I got it correct. I thought I must be wrong because the answer was so easy. It took me less than 60 seconds to see this pattern.

9. the 14,15,,16 numbers are: 131, 150 170

10. Indeed.
The sequence reads 1 3 7 12 18 26 35 45...
The differences are 2 4 5 6 8 9 10...

The point is every number not in the sequence itself appears among the differences. This property (together with the fact that both the sequence and the sequence of first differences are increasing) defines the sequence!