Exercise 28 Page 319

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

Recursive : Explicit : a_{n} = r \cdot a_{n - 1}; a_{1} = a_{1} a_{n} = a_{1} (r)^{n - 1}

In these formulas,

r

is the common ratio and

a_{1}

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

r

and the value of the first term

a_{1} .

a_{n} = - 5 a_{n - 1}; a_{1} = 5

We can see that

- 5

is the common ratio and the first term is

5 .

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

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

$a_{1} = 5$ , $r = - 5$

a_{n} = 5 (- 5)^{n - 1}