Question 1177145
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Technically, the problem as stated is invalid; ANY next number would form a valid sequence.<br>
However, there is a clear pattern in the numbers shown: the differences between successive term from an arithmetic sequence.<br>
That means the differences between those differences (mathematically, the second differences of the given sequence) are constant; in fact they are all -2.<br>
We can display all of this in an array like this:<br><pre>
   20  27  32  35  36  the given terms of the sequence
      7   5   3   1    the first differences
       -2  -2  -2      the second differences<br></pre>
A constant second difference means the sequence can be formed by a polynomial of degree 2:<br>
{{{t(n) = an^2+bn+c}}}<br>
Use that general quadratic equation with the fact that t(1)=20, t(2)=27, and t(3)=32 to write and solve a system of three equations in a, b, and c.<br>
That will give you what you are looking for -- a quadratic equation that determines the n-th term in the sequence.<br>