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What is the next number in the sequence 19, 10, 11, 18, 38,?

The options are

  1. 97.5
  2. 110
  3. 115
  4. 124.5
  5. 99.5

I got an image from a book that had this question. I can't solve it. Any hints?

question number 140 in the image

Question number 140 in the image. But I don't know from which book it is. Received it in a WhatsApp group

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  • $\begingroup$ The previous question looks like x/3-1, if this can help $\endgroup$ – Fabich Jun 21 '16 at 18:46
  • $\begingroup$ I don't know why people are down voting it. I was just curious to know the answer. That's why I uploaded the image to show that it's a genuine question published in some book and not made up on my own randomly $\endgroup$ – idpd15 Jun 21 '16 at 18:50
  • $\begingroup$ Many of these types of questions get added sporadically, people tend to turn against them as they can be somewhat arbitrary in nature. Personally I kind of like hunting for the Ah Ha! moment where you are quite sure you are right rather than a sarcastic 72nd degree polynomial. $\endgroup$ – Going hamateur Jun 21 '16 at 20:34
  • $\begingroup$ Vote of approval for 2 main reasons: 1) It's multiple choice, a reasonable way to present a short sequence. 2) Automated sequence solvers get it wrong. $\endgroup$ – humn Jun 22 '16 at 0:55
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I think the answer here is:

(1) 97.5

Because the pattern is:

19 * 0.5 + 0.5 = 10
10 * 1 + 1 = 11
11 * 1.5 + 1.5 = 18
18 * 2 + 2 = 38
38 * 2.5 + 2.5 = 97.5
Each time you multiply by then add a value increased by 0.5.
I know there's a more mathematical way to express this, but it's been a long time since I studied this.

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    $\begingroup$ This certainly seems to be in line with the question before in the book. $\endgroup$ – BmyGuest Jun 21 '16 at 22:18
  • $\begingroup$ ^vote with a note: Answer would be even better if it mentioned how you thought of it. $\endgroup$ – humn Jun 22 '16 at 0:57
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    $\begingroup$ "I know there's a more mathematical way to express this"- there is, but it's too mathematical: $a_n=2^{-n}\left( 19\cdot n!+ne ^2\Gamma(n,2)\right).$ The above explanation is much more intuitive and satisfying. $\endgroup$ – Ankoganit Jun 22 '16 at 4:54
  • $\begingroup$ Thanks. I wasn't able to solve 141 also earlier but after your answer i was able to do that also! $\endgroup$ – idpd15 Jun 22 '16 at 6:25
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For any of these types of questions, there really is no correct answer. What you're doing is extrapolating, which is a bad way of looking at the values you already have (speaking from a statistical standpoint).

From a mathematical standpoint, you could try to form a polynomial model to fit the values you currently have, and extrapolate the next value from it.

Or the next value could just be obtained from a visual or mental pattern.

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    $\begingroup$ This question was in a quantitative ability book, so i really think there should be some mathematical pattern behind it or some relation between the numbers themselves $\endgroup$ – idpd15 Jun 21 '16 at 16:00
  • $\begingroup$ I can always find a function which gives the stated values and then pi/2 as the next value. I sometimes answer it to such questions, but after a while I got bored, there are so many similar questions! $\endgroup$ – vsz Jun 21 '16 at 19:01
  • $\begingroup$ In fact, using Wolfram Alpha to find interpolation polynomial and then extrapolate, I got the answer 89... $\endgroup$ – user24855 Jun 21 '16 at 22:52
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Q 140

Solution ((19×1)+1)÷2=10
      ((10×2)+2)÷2=11
      ((11×3)+3)÷2=18
      ((18×4)+4)÷2=38
      ((38×5)+5)÷2=97.5

So ans is 97.5

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  • $\begingroup$ Welcome to Puzzling! (Take the Tour!) How does your answer add to the identical ones already given? You should always look at existing answers before providing one of your own, to ensure you are not just adding a duplicate. $\endgroup$ – Rubio♦ May 11 '17 at 18:16

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