Rosen

Question

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

The answer is $f(x) = \frac{x}{1-x^3}$ ?

Answer 1

GIven ,

Generating function pattern = { 0 , 1 , 0 , 0 , 1 , 0 , 0 , 1 ,....}

So ,  a₀  =  0

        a₁  =  1

        a₂  =  0    and so on..

We know ,

Generating function f(x)   =   Σ a_r . x^r   where this is an infinite series with r starting from 0..

So                         f(x)   = 0.x⁰   + 1.x¹  + 0.x²  + 0.x³  + 1.x⁴  + .................

                                     = x(1 + x³ + x⁶ + ...................)

                                     = x(1 / (1 - x³ ) )  [  As in the given G.P. , first term = 1 and common ratio is x and this is infinite G.P. ]

                                     = x / (1 - x³)

Hence                     f(x)  = x / (1 - x³)

Comments

@Habibkhan...

series is given as 0,1,0,0,1,0,0,1.....means we know just first 8 terms, now

what if series is like this 0,1,0,0,1,0,0,1,1,1,1,1,1,1.........

now genrating function will be,

f(x)= x + x⁴ + $\frac{x^{7}}{1-x}$

how are you deciding terms after a₇...it could be anything, and there could be different type of series..

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That way the pattern can be any..But u have to decide one..So u need to see the existing pattern na..