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November 11th, 2015, 02:50 AM   #1
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Find the generating function for the following sequence

Hello

I need to find generating function for sequence :

1,1,1,1,1,1,0,0,0,0 ....

$\displaystyle G(x) = 1 + x + x^2 + x^3 + x^4 + x^5 + 0x^6 + 0x^7 + 0x^8 + 0x^9 ..... = 1 + x + x^2 + x^3 + x^4 + x^5$

and then $\displaystyle G(x) = \frac{1 - x^6}{1 - x}$

Why is There $\displaystyle 1 - x^6$?

Can somebody help me to understand that?
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November 11th, 2015, 11:07 AM   #2
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In general, $\displaystyle a^n- b^n= (a- b)(a^{n-1}+ a^{n-2}b+ a^{n-3}b^2+ \cdot\cdot\cdot+ a^2b^{n-2}+ ab^{n-2}+ b^{n-1})$. That can be proved, for example, by induction on n.

In particular, with a= 1, b= x, and n= 6, $\displaystyle 1- x^6= (1- x)(1+ x+ x^2+ x^3+ x^.4+ x^5)$.
Thanks from topsquark and Lambert1

Last edited by Country Boy; November 11th, 2015 at 11:10 AM.
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