<PRE><B>Question 1198225</B>
&lt;br&gt;
The second sentence in your post demonstrates that in fact it is NEVER possible to come up with a definitively correct answer to ANY problem like this.&lt;br&gt;
In any such problem, there might be an observable pattern that leads to a POSSIBLE  answer to the problem; but any answer you come up with might not be &quot;right&quot;.&lt;br&gt;
Regardless of how obvious a pattern seems, in any problem like this, ANY next term in the given sequence will form a valid sequence.&lt;br&gt;
So whether a pattern is found through observation only or through formal mathematics, you can always only GUESS what the answers are.&lt;br&gt;
Having said that, let&#39;s look at your two examples.&lt;br&gt;
(a) 1, 9, 3, 27, 9, 81, 27, ____&lt;br&gt;
These are all powers of 3; so there are at least a couple of ways you can find a pattern and predict the next term.&lt;br&gt;
One way is to describe how each term in the sequence is obtained through the repeated sequence of operations &quot;multiply by 9, divide by 3&quot;:
1, times 9, gives 9
9, divided by 3, gives 3
3, times 9, gives 27
27, divided by 3, gives 9
etc....&lt;br&gt;
We could also describe the sequence as two spliced sequences:&lt;br&gt;&lt;pre&gt;
(a) 1, 9, 3, 27, 9, 81, 27, ____&lt;br&gt;
(a1) 1,   3,     9,     27, ...  (geometric sequence with common ratio 3 and first term 1)&lt;br&gt;
(a2)   9,    27,    81, ...  (geometric sequence with common ratio 3 and first term 9)&lt;br&gt;&lt;/pre&gt;
(b) -2, 0, 7, 22, 48, _____ &lt;br&gt;
I see no obvious pattern in these numbers.  One thing we can ALWAYS do with any sequence is find a polynomial that produces the given sequence.  We can use the method of finite differences to find such a polynomial.  With this sequence, the  method of finite differences gives us the following:&lt;br&gt;&lt;pre&gt;
    -2    0    7    22    48    (given terms)
        2    7   15    26       (first differences)
          5    8    11          (second differences)
            3     3             (third differences)&lt;/pre&gt;
The method of finite differences shows us a constant third difference, which means the sequence can be produced by a unique polynomial of degree 3.  Various mathematical techniques show that the unique 3rd degree polynomial that produces the given sequence is {{{t(n)=(1/2)n^3-(1/2)n^2-2}}}.&lt;br&gt;
To check this for a couple of terms....&lt;br&gt;
{{{t(3)=(1/2)27-(1/2)9-2=13.5-4.5-2=7}}}
{{{t(5)=(1/2)125-(1/2)25-2=62.5-12.5-2=48}}}&lt;br&gt;
The next term in the sequence using this polynomial is&lt;br&gt;
{{{t(6)=(1/2)216-(1/2)36-2=108-18-2=88}}}&lt;br&gt;
So IF the &quot;right&quot; answer to the problem is using this polynomial, the next number in the sequence is 88.&lt;br&gt;
Note that if we only need to find the next term in the sequence, without finding the polynomial that produces the sequence, we can simply use the array that the method of finite differences gives us by adding another constant third difference of 3 and working back up the array:&lt;br&gt;&lt;pre&gt;
    -2    0    7    22    48    88 (given terms)
        2    7   15    26    40    (first differences)
          5    8    11    14       (second differences)
            3     3     3          (third differences)&lt;/pre&gt;
This shows us that the next term in the sequence is 88, without having had to find the polynomial that produces the sequence.&lt;br&gt;
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