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The fourth term of the sequence $3, \dfrac {3}{2}, 1...$ is:

My Attempt: Checking For $A.P$

First term$(a)=3$

Second term$(t_2)=\dfrac {3}{2}$

Third term$(t_3)=1$

This will be AP if $$t_3-t_2=t_2-t_1$$ $$1-\dfrac {3}{2}=\dfrac {3}{2}-3$$ $$\dfrac {-1}{2}=\dfrac {-3}{2}$$

So, this is not AP. AND also not GP.

pi-π
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    Well, the answer is obviously $\pi$. No, really, you can always come up with an explanation why any number $X$ makes sense as the next term in any sequence. Which explanations are better and which ones worse is entirely subjective... Although the one by Shaun might be considered "natural", every number can be justified. – Dirk Apr 27 '17 at 12:59
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    I'm voting to close this question as off-topic because it belongs to the wide class of ill-posed problems like find the next number in this finite sequence. Please see https://math.stackexchange.com/questions/1790642/general-formula-for-the-1-5-19-65-211-sequence/1790666#1790666 as a reference. – Jack D'Aurizio Apr 27 '17 at 14:14
  • @JackD'Aurizio, Do you mean to say that the 3 answers here are useless. – Aryabhatta Apr 27 '17 at 14:17
  • @Aryabhatta: I mean to say that there are plenty of reasonable continuations for the sequence $3,\frac{3}{2},1$. $\frac{3}{2}$ is one of them, as the value at $x=4$ of $p(x)=\frac{x^2}{2}-3x+\frac{11}{2}$, whose values at $x=1,2,3$ match $3,\frac{3}{2},1$. – Jack D'Aurizio Apr 27 '17 at 14:21
  • @JackD'Aurizio, $P(x)=\dfrac {x^2}{2} -3x+\dfrac {11}{2}$? – Aryabhatta Apr 27 '17 at 14:23
  • Or maybe the next value is $2$, as $q(4)$ for $q(x)=\frac{1}{12}(60-25x+x^3)$. You may assume that the next number is a number of your choice and come after with a reasonable explanation for that. – Jack D'Aurizio Apr 27 '17 at 14:23
  • @JackD'Aurizio, I find Shaun answer quite straightforward. – Aryabhatta Apr 27 '17 at 14:26
  • @Aryabhatta: so? Shaun's is a reasonable continuation, I am simply stating it is not the only one, and there is no way for declaring a continuation "more natural" than another one. – Jack D'Aurizio Apr 27 '17 at 14:29

3 Answers3

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Maybe the sequence is $a_n=\frac3n$, so the next term is $\frac34$.

But it could be anything using an interpolating polynomial: http://m.wolframalpha.com/input/?i=interpolating+polynomial+%7B3%2C+3%2F2%2C+1%2C+69%7D&x=0&y=0.

Shaun
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  • Well with just three terms its hard to guess, the difference sequence can also be in AP with common difference $-2$ –  Apr 27 '17 at 12:59
  • @Shaun, I haven't read about polynomial interpolation tii! – pi-π Apr 27 '17 at 13:05
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I think it is $3/4,$ since we have $3/1, 3/2, 3/3, \cdots$

Ken
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Since we have only three terms of the sequence ..we have to assume that the sequence must follow a general pattern. I am assuming the pattern to be an^2+bn+c If we put n=1 , n=2 and n= 3 and equate it with its respective terms we get the sequence follows the general pattern of n^2/2-3n+11/2 .if we put n=4 in this equation.. Its 4th term comes out to be 3/2. But since I assumed the pattern it may not be right

Lakshya Gupta
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