Answer:
[tex]a(x)=\dfrac{1}{1+2x}[/tex]
Step-by-step explanation:
The generating function a(x) produces a power series ...
[tex]a(x)=a_0+a_1x+a_2x^2+a_3x^3+\dots[/tex]
where the coefficients are the elements of the given sequence.
We observe that the given sequence has the recurrence relation ...
[tex]a_0=1;a_n=-2a_{n-1} \quad\text{for n ${data-answer}gt;$ 0}[/tex]
This can be rearranged to ...
[tex]a_n+2a_{n-1}=0[/tex]
We can formulate this in terms of a(x) as follows, then solve for a(x).
[tex]\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}[/tex]
The generating function is ...
a(x) = 1/(1+2x)
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.
