Question 680379
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<pre>

I'm sure you meant to say:

"What <b><i>are</i></b> the next 4 numbers in this pattern?"  I don't care if this is math; grammar, spelling, and punctuation still count.

Use the method of common differences.

 1     5    13   26    45    71
  \  /  \  /  \ /  \  /  \  /
    4     8    13   19    26
     \  /  \  /  \ /  \  /
       4     5     6    7
        \  /  \  /  \  /
          1     1     1

It took 3 iterations of taking the differences to find where the differences are common.  Hence, the series is modeled by a 3rd degree polynomial.

*[tex \LARGE \ \ \ \ \ \ \ \ \ \ S(x)\ =\ ax^3\ +\ bx^2\ +\ cx\ +\ d]

Since *[tex \LARGE S(1)\ =\ 1]:

*[tex \LARGE \ \ \ \ \ \ \ \ \ \ a\ +\ b\ +\ c\ +\ d\ =\ 1]

Since *[tex \LARGE S(2)\ =\ 5]:

*[tex \LARGE \ \ \ \ \ \ \ \ \ \ 8a\ +\ 4b\ +\ 2c\ +\ d\ =\ 5]

Likewise

*[tex \LARGE \ \ \ \ \ \ \ \ \ \ 27a\ +\ 9b\ +\ 3c\ +\ d\ =\ 13]

and

*[tex \LARGE \ \ \ \ \ \ \ \ \ \ 64a\ +\ 16b\ +\ 4c\ +\ d\ =\ 26]

Solve the 4X4 system for the ordered quadruple *[tex \LARGE \left<a,\,b,\,c,\,d\right>]

Once you have the values of a, b, c, and d, you can write the function that results in the *[tex \LARGE x^{th}] term by plugging those values into:

*[tex \LARGE \ \ \ \ \ \ \ \ \ \ S(x)\ =\ ax^3\ +\ bx^2\ +\ cx\ +\ d]

From there it is a simple matter of calculating the values of *[tex \LARGE S(7)], *[tex \LARGE S(8)], *[tex \LARGE S(9)], and *[tex \LARGE S(10)]

</pre>
John
*[tex \LARGE e^{i\pi}\ +\ 1\ =\ 0]
<font face="Math1" size="+2">Egw to Beta kai to Sigma</font>
My calculator said it, I believe it, that settles it
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