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How Do Arithmetic Sequences Work?

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A sequence is a set of numbers in a specific order. They can be written as a list of number separated by commas, or they can be drawn as figures. Sequences can be divided into two main types:

1.: Those that increase by a constant number,
2.: Those that don’t increase by a constant number.

Sequences that increase by a constant number has their own cool name, which is arithmetic sequences. When you figure out what’s going on, you will be able to find an expression for any term in an arithmetic sequence. The sequence

4, 7, 10, 13 \dots

is arithmetic, as each term increases by $3$ . In the sequence

1, 3, 6, 10, 15, 21, \dots,

each term does not increase by the same number, meaning it’s not arithmetic. It does follow a pattern, as I’m sure you can tell, but it doesn’t increase by the same number every time.

If you know that the sequence is arithmetic, you can follow this recipe to make an expression for it. We often say that we can find an expression $f_{n}$ for the $n$ th number in a sequence. That just means that you need to make an expression that helps you find the next number in the sequence.

Rule

Making an Expression for an Arithmetic Sequence

1.: Find out how much the sequence increase by. This is the common difference of the sequence, which we call $d$ .
2.: Find the first number of the sequence, $f_{1}$ . Then subtract the difference from the first number to find your constant term $b$ , $f_{1} - d = b$ .
3.: Write the expression as $f_{n} = d n + b$ .

Let’s use this recipe on the sequence above.

Example 1

The numbers
$4, 7, 10, 13, \dots$

could come from the following figure:

What’s the expression for $f_{n}$ ?

You use the recipe above as you can see that the sequence increases by the same number all the time.

1.: You find that the sequence increases by $7 - 4 = 3$ every time. That makes the common difference $3$ .
2.: You take the first number of the sequence, $4$ , and subtract the difference to find the constant term. $b = 4 - 3 = 1$ .
3.: That means the expression for this sequence is $f_{n} = 3 n + 1$ .

This expression should function in a way such that when you insert

1

for

n

you should get

4

, as

4

is the first number in the sequence. If you insert

n = 2

you should get

7

, as

7

is the second number in the sequence. You can test the expression for

3

and

4

\begin{array}{l} f_{3} & = 3 \cdot 3 + 1 = 9 + 1 = 10 \\ f_{4} & = 3 \cdot 4 + 1 = 12 + 1 = 13 \end{array}

You can see that the expression works the way you wanted it to, and you can use this expression to find much larger numbers in the sequence. For example, the number at $n = 100$ is:

f_{100} = 3 \cdot 100 + 1 = 301 .

Example 2

Find an expression for this sequence:

100, 95, 90, 85, 80, \dots

This sequence is decreasing instead of increasing in value, which means the common difference is negative instead of positive. You can check all the differences:

\begin{array}{l} 100 - 95 & = 5 95 - 90 = 5 \\ 90 - 85 & = 5 85 - 80 = 5 \end{array}

As the sequence is decreasing by $5$ between each term, the difference is $- 5$ , and the sequence is arithmetic. The first term is $100$ . To find the constant term, you take the first term and subtract the difference. Stay alert now, because the difference is negative!

100 - (- 5) = 100 + 5 = 105 .

This makes the constant term $b = 105$ , making the expression

f_{n} = - 5 n + 105

f_{n} = 105 - 5 n .

You can test it for a couple of numbers:

Term $2$ :

f_{2} = 105 - 5 \cdot 2 = 105 - 10 = 95 .

Term $3$ :

f_{3} = 105 - 5 \cdot 3 = 105 - 15 = 90 .

Can you find term $7$ ? Just insert $n = 7$ and you’ll get

f (7) = 105 - 5 \cdot 7 = 70 .

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