Question 1203845
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There is no mathematics to this; it is purely entertainment.<br>
ANY number in the missing spot makes a valid sequence.<br>
If there is no information given about what kind of sequence it is, then it is only a guessing game.<br>
Spend as much (or, better yet, as little!) time as you want trying to find a pattern that predicts the missing number -- knowing that any "answer" you come up with might not be "right".<br>
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Since you have received several responses saying that it's not possible to know the answer, I will add to my response to show that there is a formal mathematical way to find ONE POSSIBLE answer to the problem.<br>
The problem shows terms 1, 2, 3, 4, and 6 of a sequence.  We can find a solution using formal mathematics if we assume that the sequence is generated by a polynomial function.  If we do that, then we are looking for a polynomial f(x) for which<br>
f(1)=20; f(2)=32; f(3)=47; f(4)=57; and f(6)=80<br>
5 known function values can be fitted with a unique polynomial of degree 4, so we are looking for a function<br>
{{{f(x)=ax^4+bx^3+cx^2+dx+e}}}<br>
that has the 5 given function values.<br>
We can use matrices (e.g., on a graphing calculator like a TI-83 or TI-84) to find the polynomial function that generates the given function values.  Doing that gives us the following polynomial:<br>
{{{f(x)=(5/12)x^4-(11/2)x^3+(289/12)x^2-28x+29}}}<br>
We can then find the missing number in the sequence by evaluating f(5), which turns out to be 64.<br>
So ONE POSSIBLE answer to the problem, using a formal mathematical process is 64.<br>
We can also find that answer, without finding the polynomial that generates the sequence, using the method of finite differences.  In a polynomial of degree 4, the 4th differences are constant.<br>
So we can call the missing term x and find the 4th differences and set them equal to find the missing term.<br><pre>

   20   32     47       57           x        80   given terms
     12   15       10         x-57       80-x     1st differences
        3     -5       x-67        137-2x        2nd differences
          -8      x-62       204-3x             3rd differences
             x-54     266-4x                   4th differences</pre>
The 4th differences must be the same:<br>
{{{x-54=266-4x}}}
{{{5x=320}}}
{{{x=64}}}<br>