Question 1139208
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The answer from the other tutor is one POSSIBLE answer that is obtained by logical reasoning.<br>
But there is no single "right" answer to ANY problem like this, because different logical reasoning might produce a different next number.<br>
And, in the end, you could put ANY number you want next and it would be a perfectly valid sequence.<br>
For any problem like this to have any meaning, information must be provided indicating what kind of sequence we are looking for.<br>
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Note one formal mathematical method that can theoretically be used to solve any problem like this is to assume the numbers are the values of a polynomial evaluated for x=1 to x=5.<br>
5 points can be fit with a polynomial of degree 4; you could use the given values to find the polynomial that produces these numbers and then evaluate the polynomial for x=6 to find an answer that results from formal mathematics.<br>
If you only need to find the next number produced by that polynomial, you can use the method of finite differences to find it, as follows:<br><pre>
    3   -6   12    4   20    <--  the given sequence of numbers
      -9   18   -8   16      <--  the differences between terms ("first differences")
         27   -26  24        <--  second differences
           -53   50          <--  third differences
              153            <--  fourth difference<br></pre>
To find the next term in the sequence assuming a polynomial of degree 4, repeat the "153" in the row of fourth difference and work back up the array to find the next number in the sequence:<br><pre>
    3   -6   12    4   20   263
      -9   18   -8   16   243
         27   -26  24  227
           -53   50  203
              153  153<br></pre>
The next number in the sequence -- assuming the sequence is produced by a polynomial of degree 4 -- is 263.