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

Recursive : Explicit : a_{n} = a_{n - 1} + d; a_{1} = a_{1} a_{n} = a_{1} + (n - 1) d

In both of these rules,

d

a_{1}

is the first term. Let's start by substituting

1

for

n

in the given rule in order to find the value of

a_{1} .

a_{n} = 9 - 5 n

$n = 1$

a_{1} = 9 - 5 (1)

Identity Property of Multiplication

a_{1} = 9 - 5

Subtract term

a_{1} = 4

We know that

a_{1} = 4,

so now we need to identify the value of

d .

Let's take a look at the given rule.

a_{n} = 9 - 5 n

Notice that the coefficient of

n

is

- 5,

so we know that the terms are decreasing by

5

each time. For the

n^{th}

term, the number of times it has decreased in relation to

a_{1}

is given by the value of

n .

Therefore, the common difference

d

is

- 5 .

Now we have enough information to form a recursive rule for our sequence.

a_{n} = a_{1} = a_{n - 1} + (- 5); 4

Therefore, the recursive rule for our sequence is

a_{1} = 4,

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

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