Question

1. a sequence is defined by ( f(0)=4 ) and ( f(n)=2cdot f(n - 1) ). write the explicit rule for this function.
2. a sequence is defined by ( f(0)= - 1 ) and ( f(n)=f(n - 1)-3 ). write the explicit rule for this function.
3. write the recursive and explicit rule for the function graphed below.

explicit rule:

Explanation:

4.

Step1: Identify the type of sequence

This is a geometric sequence since $f(n)=2\cdot f(n - 1)$ with $a = f(0)=4$ and common - ratio $r = 2$.
The general formula for a geometric sequence is $f(n)=a\cdot r^{n}$.

Step2: Substitute values into the formula

Substitute $a = 4$ and $r = 2$ into the formula, we get $f(n)=4\cdot2^{n}=2^{2}\cdot2^{n}=2^{n + 2}$.

Step1: Identify the type of sequence

This is an arithmetic sequence since $f(n)=f(n - 1)-3$ with $a=f(0)=-1$ and common - difference $d=-3$.
The general formula for an arithmetic sequence is $f(n)=a+(n)d$.

Step2: Substitute values into the formula

Substitute $a=-1$ and $d = - 3$ into the formula, we get $f(n)=-1+(n)(-3)=-1-3n$.

Step1: Determine the type of function

The points $(0,100),(1,50),(2,25),(3,12.5),(4,6.25)$ suggest an exponential function of the form $y = a\cdot b^{x}$.
When $x = 0$, $y=a\cdot b^{0}=a$. Since when $x = 0,y = 100$, then $a = 100$.
To find $b$, we know that when $x = 1,y=50$. Substitute into $y=a\cdot b^{x}$, we have $50=100\cdot b^{1}$, so $b=\frac{1}{2}$.
The explicit rule is $f(x)=100\cdot(\frac{1}{2})^{x}$.
For the recursive rule, we know that $f(n)=b\cdot f(n - 1)$. Here $b=\frac{1}{2}$, and $f(0)=100$. So the recursive rule is $f(0)=100,f(n)=\frac{1}{2}f(n - 1)$ for $n\geq1$.

Answer:

$f(n)=2^{n + 2}$

5.