What number comes next in the sequence $7, 16, 8, 27, 9,...$?

Question

What number comes next in this series?
$$7, 16, 8, 27, 9,...$$

I thought it was $38$, but I'm wrong.

It is a multiple choice, and options are $27, 10, 40, 37$.

Don't worry - I'm not cheating on anything, but helping my daughter with homework.
And I can't help her since I can't figure it out myself!

$42$ is as good as anything. There's no single answer, I'm afraid. The only sequences I know that you can say for certain what the next number is are $640$, $231$, $100$, $91$, $\dots$ and $5$, $10$, $20$, $30$, $36$, $\dots$. — egreg, Feb 12 '16 at 18:33
$40$. I am sort of joking. There is a semi-plausible pattern that does yield $40$, but the sample size is too small. — André Nicolas, Feb 12 '16 at 18:35
I'd have gone for $2^4 = 6$ or $4^2 = 16$ (both are equal), on the grounds that the terms are $7,8,9,\dots$ interwoven with $2^4, 3^3, 4^2, \dots$ or $4^2, 3^3, 2^4, \dots$ — Patrick Stevens, Feb 12 '16 at 18:37
Oh I see how you got 38 I think. You looked at the digits of the even entries. And each digit increased each time over the even entries. — randomgirl, Feb 12 '16 at 18:40
Like André, I came up with $40$: $16=2\cdot7+2$, $27=3\cdot8+3$, $40=4\cdot9+4$. — Brian M. Scott, Feb 12 '16 at 18:45
I guess I could have just said you did +11 each time for the even tries. This definitely seems like a possible answer 38. — randomgirl, Feb 12 '16 at 18:45
I know this is my third comment, but does it really say what number comes next in the series? Or did it say sequence? — randomgirl, Feb 12 '16 at 18:47
Hm, I see a prime, a square, a cube, a cube, a square, so if I'm expecting a prime next, what would be wrong with it? I guess there is a countable infinity of ways to look at this. — Gyro Gearloose, Feb 12 '16 at 19:35
3 votes to close and 4 up votes, we're surely a divided audience. — YoTengoUnLCD, Feb 13 '16 at 00:36
Too bad multiple choice did not include 38 a(n) = 1/4((10n - 3)(-1)^n + 12n + 23) for n>=1 7, 16, 8, 27, 9, 38, 10, 49, 11, 60, 12, 71, 13, 82, 14, 93, 15, 104, 16, ... — Alex, Feb 14 '16 at 22:09

score 8 · Answer 1 · answered Feb 12 '16 at 18:37

8

To me, it's $40$.

Indeed

$$16 = 2\cdot 8$$ $$27 = 3\cdot 9$$

thence

$$40 = 4\cdot 10$$

The even terms of the series may follow this path, so a possible series could be

$$7, 16, 8, 27, 9, 40, 10, 55, 11, 72, 12, \cdots$$

answered Feb 12 '16 at 18:37

Enrico M.

26,114

score 8 · Answer 2 · answered Feb 12 '16 at 18:38

8

One possibility is that the $1$st, $3$rd, $5$th, and so on, numbers are just increasing by one, so $$7,\_,8,\_,9,\_,10,\_,...$$

Now notice that $16=2 \cdot 8$, and $27=3 \cdot 9$. Therefore it would be reasonable to assume the next number to be $40=4 \cdot 10$.

answered Feb 12 '16 at 18:38

Bobson Dugnutt

10,672

This was also my train of thought. – Alenanno Feb 12 '16 at 19:01

Dietrich Burde · Answer 3 · 2016-02-12T19:20:45.920

5

The next term is again $27$; then we had a palindromic sequence $$ 7,16,8,27,9,27,8,16,7,\ldots, $$ repeating like this.

edited Feb 12 '16 at 19:20

answered Feb 12 '16 at 18:40

Dietrich Burde

130,978

score 4 · Answer 4 · answered Feb 12 '16 at 19:48

4

$40$, because the sequence is $n+7$ and $n^2+10n+16$ interleaved.

$37$, because the sequence is $n+7$ and $-n^2/2+23n/2+16$ interleaved.

Use imagination and you come with equally good excuses for the other two options.

answered Feb 12 '16 at 19:48

yo'

4,506

score 2 · Answer 5 · answered Feb 12 '16 at 19:12

Another view, also giving $40$ as the next term: $2n+2;\frac{n}{2};3n+3;\frac{n}{3};4n+4;\frac{n}{4}\cdots$ \begin{align}\text{initial value}&=7\\ 7*2+2&=16\\ \frac{16}{2}&=8\\ 8*3+3&=27\\ \frac{27}{3}&=9\\ 9*4+4&=40\\ \frac{40}{4}&=10 \end{align} Or: $2(n+1);\frac{n}{2};3(n+1);\frac{n}{3};4(n+1);\frac{n}{4}\cdots$ \begin{align}\text{initial value}&=7\\ 2(7+1)&=16\\ \frac{16}{2}&=8\\ 3(8+1)&=27\\ \frac{27}{3}&=9\\ 4(9+1)&=40\\ \frac{40}{4}&=10 \end{align}

What number comes next in the sequence $7, 16, 8, 27, 9,...$?

5 Answers5