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Find the generating functions for the following sequence (i) 0, 0, 0, 1, 2, 3, 4, 5, 6, 7,............................. (ii) 6, -6, 6, -6, 6, -6, 6,...................
1 Answer
written 7.4 years ago by |
0, 0, 0, 1, 2, 3, 4, 5, 6, 7,……………..
Assume the generating function
$f(x)=a_0+a_1x+a_2 x^2+a_3 x^3+……$
But, the given sequence is {0, 0, 0, 1, 2, 3, 4……}. Using this sequence, the expression above becomes
f(x)$=0 +0x+0x^2+1x^3+2x^4+3x^5+4x^6+…… \\ =1x^3+2x^4+3x^5+4x^6+…… \\ =x^3 (1+2x+3x^2+4x^3+…….) \\ =x^3 (1-x)^{-2}$
Accordingly, $f(x) = x^3 …