## 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

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

What is the next number in this sequence:

1,4,9,16, …

Note: the answer is NOT 25

January 2nd, 2011 at 3:17 am

is it… nm. idk

January 2nd, 2011 at 6:11 am

144

January 2nd, 2011 at 8:27 am

4

January 2nd, 2011 at 8:54 am

27? Increasing by primes?

January 2nd, 2011 at 9:19 am

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!

January 2nd, 2011 at 9:55 am

49

January 2nd, 2011 at 10:24 am

26

January 2nd, 2011 at 10:24 am

23!

January 2nd, 2011 at 10:33 am

maybe 27

January 2nd, 2011 at 11:02 am

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.

January 2nd, 2011 at 11:04 am

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

January 2nd, 2011 at 11:08 am

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

January 2nd, 2011 at 11:24 am

the next number is 30

January 2nd, 2011 at 2:11 pm

So far the only correct answer has been given by Bhargav

( 49 ). How about supplying the next number in the sequence.

January 2nd, 2011 at 7:20 pm

27 is the next number

January 2nd, 2011 at 7:56 pm

is it 25 rofl

January 2nd, 2011 at 7:57 pm

no wait its 27, my bad…

January 2nd, 2011 at 10:18 pm

the next number is 27

January 2nd, 2011 at 10:22 pm

the next number is 81

January 3rd, 2011 at 8:28 am

Your Comment…

January 3rd, 2011 at 8:29 am

it is 81

January 3rd, 2011 at 3:01 pm

121 is after 49

January 3rd, 2011 at 3:04 pm

Scratch that. Never as easy as it first appears…

January 4th, 2011 at 12:21 am

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)

January 4th, 2011 at 12:47 am

24

January 4th, 2011 at 4:54 pm

well. i think he gave us the answer.

the next number

is

NOT 25

January 4th, 2011 at 4:55 pm

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.

January 4th, 2011 at 5:14 pm

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 )

January 4th, 2011 at 7:19 pm

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.

January 4th, 2011 at 8:19 pm

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,…)

January 4th, 2011 at 9:33 pm

I did’t expect it to look so hairy. I wrote it as:

(n-1)(n-2)(n-3)(n-4)+n^2

January 4th, 2011 at 9:44 pm

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

January 5th, 2011 at 1:52 am

my answer is 24

January 5th, 2011 at 9:21 am

its 21,not 25.

January 9th, 2011 at 10:29 pm

1 (+3) 4 (+5) 9 (+7) 16 (+11) = 27

Find it by adding the next consecutive prime number.

January 9th, 2011 at 10:31 pm

Simply put, the pattern is adding the next consecutive prime number.

1 (+3) 4 (+5) 9 (+7) 16 (+11) = 27

October 9th, 2014 at 8:01 pm

1, 4, 9, 16,…simply 1 + 4 + 4 = 9

4 + 9 + 4 = 16

9 + 16+ 4 = 29>>>