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How to Complete a Linear Number Pattern

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  • 0:04 Linear Number Patterns
  • 1:00 The Formula
  • 1:35 Using the Formula
  • 2:37 Example
  • 3:49 Lesson Summary
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Lesson Transcript
Instructor
Yuanxin (Amy) Yang Alcocer

Amy has a master's degree in secondary education and has taught math at a public charter high school.

Expert Contributor
Laura Pennington

Laura received her Master's degree in Pure Mathematics from Michigan State University. She has 15 years of experience teaching collegiate mathematics at various institutions.

After completing this lesson, you'll know how to write the equation for a given linear number pattern. You'll learn what to look for in the pattern and which formula to use to help you find the equation.

Linear Number Patterns

Did you know that when you're counting in the game of hide and seek that you are actually using a linear number pattern? That's right. The sequence of one, two, three, and so on is a linear number pattern. A linear number pattern is a sequence of numbers whose difference between all the terms is the same. When you count, the difference between each successive number is 1. You start with 1, then you add 1 to get 2. Then you add another 1 to get 3. And you keep repeating this addition of 1 process to get the rest of the terms.

Here are some other examples of linear number patterns:

  • Starting with three and a difference of two: 3, 5, 7, 9, …
  • Starting with five and a difference of three: 5, 8, 11, 14, …

Sometimes, you need to answer problems that ask you to write an equation for the linear number pattern. For example, you might see this problem:

  • Complete the linear number pattern by writing the equation for the pattern that begins with 2 and has a difference of 2

The Formula

To solve this problem, you'll need the formula for linear number patterns:

  • an = dn - c

The an stands for the nth term of the pattern. So a1 is the first term and a2 is the second term of the pattern. The d is the difference between each of the terms. The n stands for the term you're calculating. If you're calculating the third term, then n is 3. And c is a constant that you'll need to calculate.

For example, the linear number pattern of 1, 3, 5, 7, 9, … has an equation of:

an = 2n - 1.

Using the Formula

Now, let's see how you can use this formula to help you solve a problem:

  • Complete the linear number pattern by writing the equation for the pattern that begins with 2 and has a difference of 2

The problem tells you that the first term is two and the difference between the terms is two. So d = 2 and a1 = 2.

Using this information, you can go ahead and find c. You'll then have the equation for the linear number pattern. To do this, you plug in 2 for a1, 2 for d, and 1 for n. This shows that you are making the calculation for the first term.

  • a1 = d(1) - c
  • 2 = 2(1) - c

You can now solve for c:

  • 2 = 2 - c
  • 2 - 2 = 2 - 2 - c
  • 0 = c

You subtracted 2 from both sides to isolate c and you find that c equals 0.

Now complete the equation for this linear number pattern:

  • an = 2n

Remember, when you write the final form of the equation, you leave the an as it is and fill in the values for d and c.

Example

Let's try another example:

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Additional Activities

Real World Applications of Linear Number Patterns


Reminders

  • A linear number pattern is a list of numbers in which the difference between each number in the list is the same.
  • The formula for the nth term of a linear number pattern, denoted an, is an = dn - c, where d is the common difference in the linear pattern and c is a constant number.


Real World Applications

1) Jared is participating in a fitness walking challenge. The challenge is to walk 15 miles per week for a year. Find a formula for the total number of miles that Jared has walked at the end of the nth week of the challenge. Use this formula to determine how many miles that Jared will have walked at the end of one year (Hint: There are 52 weeks in a year).


2) Connie is saving for a bike that costs $90. She starts with $30 in a jar and adds $20 to the jar each week. Find a formula for the amount of money in the jar after n weeks. Use this formula to determine how many weeks it will take Connie to save enough money to buy the bike.


3) A large rock formation is eroding in such a way that it is losing three inches in height each year. If the rock formation has an original height of 53 feet (636 inches), find a formula for the height in inches of the rock formation after n years. Use the formula to determine how many years it will take for the rock formation to disappear completely (have a height of zero inches).


Solutions

1) The formula for the total number of miles that Jared has walked at the end of the nth week of the challenge, denoted as an, is an = 15n. At the end of one year, Jared will have walked 15(52) = 780 miles.


2) The formula for the amount of money in the jar after n weeks, denoted as an, is:

  • an = 20n + 10
  • 90 = 20n + 10
  • 80 = 20n
  • 4 = n

So it will take Connie four weeks to save enough money for the bike.


3) The formula for the height in inches of the rock formation after n years, denoted as an, is:

  • an = -3n + 639
  • 0 = -3n + 639
  • -639 = -3n
  • 213 = n

So the time that it will take for the rock formation to disappear completely is 213 years.

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