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## A Difficult Sequence

Posted by Al Gelman on January 2, 2011 – 12:30 am

This is a tough one.
What is the next number in this sequence:
1,4,9,16, …
Note: the answer is NOT 25

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This post is under “Tom” and has 37 respond so far.

### 37 Responds so far- Add one»

1. 1. sarah henson Said：

is it… nm. idk

144

3. 3. Chris Said：

4

4. 4. Karl Sharman Said：

27? Increasing by primes?

5. 5. Genius Said：

Its 27!
1 + 3 is 4
4 + 5 is 9
9 + 7 is 16
and 16 + 11(which is the next prime no.) is 27!

6. 6. Bhargav Said：

49

7. 7. ahmed Said：

26

8. 8. smarty! Said：

23!

9. 9. mahmoud Said：

maybe 27

10. 10. Chris Said：

I took 1 and 4 to be seeds. Then 9 = (4-(1))², 16 = (9 -(4+1))². Then (16 – (9+4+1))² = 2² = 4. The next term would be (4 – (16+9+4+1))² = 26² = 676.

Karl’s suggestion, but odd primes, seeding with 1 =>
1+3 = 4. 4 + 5 = 9. 9 + 7 = 16. 16 + 11 = 27. So 27 + 13 = 40 would be next. That works nicely too. In fact I prefer it to my guess.

11. 11. benJAMIN Said：

I agree with karl. The primes are 3, 5, 7, 11. 1 + 3 = 4 + 5 = 9 + 7 = 16 + 11 = 27. It is the only option

12. 12. benJAMIN Said：

Now I see Chirs’s suggestion and considering the fact that the title is “A Difficult Sequence” chris’s suggestion would be much more fitting to the term difficult

13. 13. khan Said：

the next number is 30

14. 14. Al Gelman Said：

So far the only correct answer has been given by Bhargav
( 49 ). How about supplying the next number in the sequence.

27 is the next number

16. 16. Megan Said：

is it 25 rofl

17. 17. Megan Said：

no wait its 27, my bad…

the next number is 27

19. 19. eva Said：

the next number is 81

20. 20. viraj Said：

it is 81

22. 22. Another Schmo Said：

121 is after 49

23. 23. Another Schmo Said：

Scratch that. Never as easy as it first appears…

24. 24. Al Gelman Said：

Here is a hint.
1,4,9,16,49,156, … , f(N) + g(N), ….

(where f(N) is some funtion of N, and g(N) is a different funtion of N)

25. 25. oliver Said：

24

26. 26. Chris Said：

well. i think he gave us the answer.
the next number
is
NOT 25

27. 27. Chris Said：

Are 409 and 904 the next two?

If so I know the equation. But that couldn’t have reasonably been determined until you had provided the 49. Even then the 156 was needed to give the required confidence that it was right – if it is.

28. 28. Chris Said：

Hi Al. You have quite a few pending posts for this blog. Although you can post, you may not be a full-blown author. You’ll have to ask Rajesh Lal to promote you (if he doesn’t do so automatically after reading this post )

29. 29. Al Gelman Said：

Hi Chris,
Yes you are correct. The next two terms are indeed 409 and 904.
I’m sorry that I didn’t provide the 5th term (49) in the original statement of the problem. I realize now that a solution was almost impossible without that at least. Figuring out this sequence was very clever, and I congratulate you.
I’m not sure I understand what you mean by “becoming a full-blown author”, as I’m quite new to this blog.

30. 30. Chris Said：

Hi Al. A full author can approve and delete posts on his own blog. At this moment you’ve got about a dozen posts that have been awaiting approval from yourself. Only Rajesh Lal (the host of this site) can grant you that status.

After getting the last two numbers, I was able to make a difference table and found that you’d used a fourth order polynomial. So I then worked the table backwards to get the 409 and 904 (as I didn’t need to know the polynomial to do that).

1,4,9,16,49,156 / 409,904
3,5,7,33,107 / 253,495
2,2,26,74 / 146,242
0,24,48 / 72,96
24,24 / 24,24

Then I used wolframalpha.com to find the polynomial:
n^4 -10*n^3 + 36*n^2 – 50*n + 24 (n = 1,2,3,…)

31. 31. Al Gelman Said：

I did’t expect it to look so hairy. I wrote it as:
(n-1)(n-2)(n-3)(n-4)+n^2

32. 32. Chris Said：

Hi Al. I had wondered why you’d written it as f(N) + g(N). Now I’ve seen that formula, I can understand why you thought that it should have been soluble with the info you’d given.

I acknowledge that was a clever way to generate your cunningly fiendish designer series. Nice one

33. 33. shantz Said：

34. 34. JZ Samson Said：

its 21,not 25.

35. 35. Shannon Said：

1 (+3) 4 (+5) 9 (+7) 16 (+11) = 27
Find it by adding the next consecutive prime number.

36. 36. Shannon Said：

Simply put, the pattern is adding the next consecutive prime number.
1 (+3) 4 (+5) 9 (+7) 16 (+11) = 27

37. 37. walter Said：

1, 4, 9, 16,…simply 1 + 4 + 4 = 9
4 + 9 + 4 = 16
9 + 16+ 4 = 29>>>

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