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November 11th, 2015, 02:50 AM  #1 
Newbie Joined: Nov 2015 From: Poland Posts: 4 Thanks: 0  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? 
November 11th, 2015, 11:07 AM  #2 
Math Team Joined: Jan 2015 From: Alabama Posts: 3,264 Thanks: 902 
In general, $\displaystyle a^n b^n= (a b)(a^{n1}+ a^{n2}b+ a^{n3}b^2+ \cdot\cdot\cdot+ a^2b^{n2}+ ab^{n2}+ b^{n1})$. That can be proved, for example, by induction on n. In particular, with a= 1, b= x, and n= 6, $\displaystyle 1 x^6= (1 x)(1+ x+ x^2+ x^3+ x^.4+ x^5)$. Last edited by Country Boy; November 11th, 2015 at 11:10 AM. 

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