Exercise 8 Page 202

Explicit equations for arithmetic sequences follow a specific format.

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

In this form,

a_{1}

is the first term in a given sequence,

d

is the common difference from one term to the next, and

a_{n}

is the

n^{th}

term in the sequence. For this exercise, the first term is

a_{1} = 4 .

Let's observe the other terms to determine the common difference

d .

4 \to + 1 5 \to + 1 6 \to + 1 7 \dots

By substituting these two values into the explicit equation and simplifying, we can find the formula for this sequence.

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

SubstituteII

$d = 1$ , $a_{1} = 4$

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

Distr

Distribute $1$

a_{n} = 4 + n - 1

SubTerm

Subtract term

a_{n} = n + 3

This equation can be used to find any term in the given sequence. To find

a_{25},

the

25^{th}

term in the sequence, we substitute

25

for

n .

a_{n} = n + 3

Substitute

$n = 25$

a_{25} = 25 + 3

AddTerms

Add terms

a_{25} = 28

The

2 5^{th}

term in the sequence is

28 .

Extra

More About Sequences

There is a lot of helpful information about arithmetic sequences available in our original course content!

Another common type of sequence is a geometric sequence, which has a common ratio instead of a common difference — the consecutive numbers are multiplied by the same number each time instead of having the same number added to them.