Exercise 43 Page 448

The explicit rule of a geometric sequence combines the information provided by the two equations of the recursive rule into a single equation.

Recursive : Explicit : a_{1} = a_{1}, a_{n} = r \cdot a_{n - 1} a_{n} = a_{1} r^{n - 1}

In these rules,

r

is the common ratio and

a_{1}

is the first term. Looking at the given recursive rule, we can identify the common ratio

r

and the value of the first term

a_{1} .

a_{1} = - 2, a_{n} = 3 a_{n - 1}

We can see that

3

is the common ratio and the first term is

- 2 .

Now we have enough information to form an explicit rule for this sequence.

a_{n} = a_{1} r^{n - 1}

$a_{1} = - 2$ , $r = 3$

a_{n} = - 2 (3)^{n - 1}