# Arithmetic sequence

Before talking about arithmetic sequence, in math, a sequence is a set of numbers that follow a pattern. We call each number in the sequence a term.

For examples, the following are sequences:

1, 4, 7, 10, 13, 16, 19, .......

70, 62, 54, 48, 40, ...............

An arithmetic sequence is a sequence where each term is found by adding or subtracting the same value from one term to the next. We call this value "common sum" or "common difference"

Looking at 1, 4, 7, 10, 13, 16, 19, ......., carefully helps us to make the following observation:

As you can see, each term is found by adding 3, a common sum to the previous term

Looking at 70, 62, 54, 46, 38, ...............carefully helps us to make the following observation:

This time, to find each term, we subtract 8, a common difference from the previous term

Many arithmetic sequences can me modeled with an algebraic expression

Here is the trick or recipe per se!

Let us try to model 1, 4, 7, 10, 13, 16, 19, .......

Let n represent any term number in the sequence
The number we add to each term is 3

The number that comes right before 1 in the sequence is -2

We can therefore model the sequence with the following formula:

3 × n + -2

Check:

When n = 1, which represents the first term, we get 3 × 1 + -2 = 3 + -2 = 1

When n = 2, which represents the second term, we get 3 × 2 + -2 = 6 + -2 = 4

Let us try to model 70, 62, 54, 46, 38, ...............

Let n represent any term number in the sequence
The number we subtract to each term is -8

The number that comes right before 70 in the sequence is 78

We can therefore model the sequence with the following formula:

-8× n + 78

Check:

When n = 1, which represents the first term, we get -8 × 1 + 78 = -8 + 78 = 70

When n = 2, which represents the second term, we get -8 × 2 + 78 = -16 + 78 = 62

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Bessy has 6 times as much money as Bob, but when each earn \$6. Bessy will have 3 times as much money as Bob. How much does each have before and after