Question 1133165
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I will copy the very good response from tutor @MathLover1 and expand on it a bit....<br>
{{{8}}},{{{18}}},{{{30}}},{{{44}}} -> find differences first

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I revised and added some to this display....<br><pre>

     8      18      30      44     (the given sequence)
        10      12      14         (the "first" differences -- differences between successive terms)
            2       2              (the "second differences -- differences between successive first differences)</pre>

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The pattern goes +{{{10}}}, +{{{12}}}, +{{{14}}}, ...

Since the second difference is always +{{{2}}}, the nth term must involve {{{n^2}}}.

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I'll add a lot here, assuming you are not familiar with what she says there.<br>
In this problem, the second differences are a constant 2, which is 2!.  In general, if the second differences are a constant 2k, then the sequence can be produced by a polynomial with leading term kx^2.  So, for example, if you have a similar problem where the constant second differences are 8, the sequence can be produced by a polynomial with leading term 4x^2.<br>
The idea continues for polynomials of higher degrees:<br>
If the polynomial has leading term kx^3, then the constant 3rd differences will be 6k (where 6 = 3!).<br>
If the polynomial has leading term kx^4, then the constant 4th differences will be 24k (where 24 = 4!).<br>
And so on....<br>

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So let's say the nth number in the sequence = {{{n^2 + an + b}}}

To find {{{b}}}, you can substitute in {{{n=0}}}, i.e. what is the "zeroth" number in the sequence? In other words, what number would have come before {{{8}}}?

This number must have been {{{0}}}, in order to continue the pattern of +{{{8}}}, +{{{10}}}, +{{{12}}}, +{{{14}}}, ...

So we now know the nth number = {{{n^2 + an}}}

Now just plug in any other value we know, to find the value of{{{ a}}}. For example, using the first number in the sequence ({{{n=1}}}), we get:
{{{1 + a = 8}}}
{{{a = 7}}}


Therefore the nth term in this sequence is: {{{n^2 + 7n }}}