Question

Find the generating function for the sequence 1,-2,4,-8, 16, ...

Answer 1

Answer:

  $a(x)=\dfrac{1}{1+2x}$

Step-by-step explanation:

The generating function a(x) produces a power series ...

  $a(x)=a_0+a_1x+a_2x^2+a_3x^3+\dots$

where the coefficients are the elements of the given sequence.

We observe that the given sequence has the recurrence relation ...

  $a_0=1;a_n=-2a_{n-1} \quad\text{for n > 0}$

This can be rearranged to ...

  $a_n+2a_{n-1}=0$

We can formulate this in terms of a(x) as follows, then solve for a(x).

$\sum\limits^{\infty}_{n=1} {a_{n}x^n} =a(x)-a_0 \quad\text{and}\\\\\sum\limits^{\infty}_{n=1} {2a_{n-1}x^n} =(2x)a(x) \quad\text{so}\\\\\sum\limits^{\infty}_{n=1} {(a_n+2a_{n-1})x^n}=0=a(x)-a_0+2xa(x)\\\\a(x)=\dfrac{a_0}{1+2x}=\dfrac{1}{1+2x}$

The generating function is ...

  a(x) = 1/(1+2x)

Answer 2

Answer:-3+double the number

Step-by-step explanation: It started off as -3 so to get the next answer you double the number and change the sign for each number.