Writing and Using Explicit Rules for Arithmetic Sequences

a

Using the graph, we can list the first five terms of the sequence. Notice that the $x$ -axis shows us $n,$ the term number. The $y$ -axis show us $a_{n},$ the value of the terms. $1st term : 2nd term : 3rd term : 4th term : 5th term : - 3 - 1 135$

b

Let's use the formula for the

n th

term of an arithmetic sequence to write an equation for the sequence.

a_{n} = a_{1} + d (n - 1)

In this equation,

a_{1}

is a first term and

d

is the common difference. From Part A, we know that the first term is

- 3 .

To find

d

we can calculate the difference between any two consecutive terms. Here we will use the fourth and fifth terms.

a_{5} - a_{4} = d \Rightarrow 5 - 3 = 2

Now we can substitute these values into the formula.

a_{n} = a_{1} + d (n - 1)

SubstituteII

a_{1} = - 3

,

d = 2

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

DistrDistribute

2

a_{n} = - 3 + 2 n - 2

SubTermSubtract term

a_{n} = 2 n - 5

The formula for the

n

th term is

a_{n} = 2 n - 5 .

c

To write the function we can use the slope-intercept form of the equation.

y = m x + b

We can take two points from the graph and substitute them into the Slope Formula. Let's use

(3, 1)

and

(4, 3) .

m = \frac{y _{2} - y _{1}}{x _{2} - x _{1}}

SubstitutePointsSubstitute

(3, 1)

&

(4, 3)

m = \frac{3 - 1}{4 - 3}

Simplify

SubTermsSubtract terms

m = \frac{2}{1}

DivByOne

\frac{a}{1} = a

m = 2

Let's now calculate

b

by substituting

m

and the point

(3, 1)

into the equation.

y = m x + b

Substitute

m = 2

y = (2) x + b

SubstituteII

x = 3

,

y = 1

1 = 2 (3) + b

Simplify

MultiplyMultiply

1 = 6 + b

SubEqn

LHS - 6 = RHS - 6

- 5 = b

RearrangeEqnRearrange equation

b = - 5

Now that we know both

m

and

b,

we can write the function.

y = 2 x - 5

Writing and Using Explicit Rules for Arithmetic Sequences 1.4 - Solution