# Special Sequences and How They Are Generated

## Special Sequences

Sequences are a string of numbers. When a sequence has a unique pattern to it, we call it a **special sequence**. You can even think of our number line, our counting numbers, as a special sequence. You learned it when you were little, so you can count - one, two, three, and so on. What is the pattern to our counting numbers? Our counting numbers have the pattern that each number is equal to the previous number plus one.

Besides our counting numbers, there are actually quite a few special sequences out there. We will talk about the more popular ones in this video lesson. It's good to know about these special sequences because you will actually see and use these numbers often in math problems.

## Triangular and Tetrahedral Sequences

We begin with triangular and tetrahedral sequences. A **triangular sequence** is a sequence that gives you the numbers needed to form a triangle. Think of yourself building triangles with marbles. You begin with one marble. To get the next triangle, you would need to add two more marbles. So now, you have a total of three marbles.

To grow your triangle again, how many marbles would you need? You would need three more marbles to form the new base of your triangle. Now you have six marbles. Your sequence is now 1, 3, 6... Do you see the pattern? To make our triangles, we add two to our first term, three to our second term, and so on. We continue our sequence by adding four to the third term to get 10, then adding five to the fourth term to 15. We now have 1, 3, 6, 10, and 15.

Our next special sequence is a **tetrahedral sequence**. This is a sequence of the number of units needed to form a tetrahedron, a triangular pyramid. Think of the triangles we just built as each layer of our tetrahedron. So, a tetrahedron of height one will have one marble. For a height of two, it will have one plus three marbles, or four marbles. For a height of three, it will have four plus six, or ten, marbles. We are essentially adding our triangular numbers up with each term. So, our series looks like 1, 4, 10, 20, etc.

## Square and Cube Sequences

Our next group of special sequences includes the square and cube sequences. The **square sequence** is a sequence in which each term is the square of the corresponding number on the number line. So, the first term is one squared. The second term is two squared; the third term is three squared. So, this sequence looks like 1, 4, 9, 16, etc.

The **cube sequence** is a sequence in which each term is the cube of the corresponding number on the number line. So, the first term here is one cubed; the second term, two cubed; the third term, three cubed; and so on. This sequence looks like 1, 8, 27, 64, etc.

## Fibonacci Sequence

The last sequence we are going to discuss is the famous **Fibonacci sequence**. This sequence is a sequence in which each number is the sum of the previous two numbers. So, we start with one for the first term and one also for the second term (since that one is the sum of the previous two terms and we only have a one before it). The third term is then one plus one, or two. The fourth term is one plus two, or three. The fifth term is two plus three, or five. So, the sequence looks like 1, 1, 2, 3, 5, 8, etc.

This is a famous sequence because things in nature actually follow this sequence. Count the number of petals on a flower, and you will find that most flowers will have a number in this sequence. Also, if you look at plants that spiral, you will see that the number of petals at each level corresponds to the Fibonacci sequence.

## Lesson Summary

And there we have it. Now, let's review. A sequence with a unique pattern to it is called a **special sequence**. A **triangular sequence** is a sequence that gives you the numbers needed to form a triangle. It has a sequence that begins with 1, 3, 6, 10, etc. A **tetrahedral sequence** is a sequence of the number of units needed to form a tetrahedron, a triangular pyramid. Its sequence begins with 1, 4, 10, 20, etc.

A **square sequence** is a sequence in which each term is the square of the corresponding number on the number line. It begins with 1, 4, 9, 16, etc. A **cube sequence** is a sequence in which each term is the cube of the corresponding number on the number line. It begins with 1, 8, 27, 64, etc. And lastly, the famous **Fibonacci sequence** is a sequence in which each number is the sum of the previous two numbers. It begins with 1, 1, 2, 3, 5, etc.

## Learning Outcomes

You should have the ability to do the following after watching this video lesson:

- Define special sequence
- Describe triangular, tetrahedral, square and cube sequences
- Explain what the Fibonacci sequence is and why it is important

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## Practice Sequences

#### Problem 1:

Find a formula for the general term of the sequence 1/2, 1/4, 1/8, 1/16.., assuming the pattern of the first few terms continues.

#### Problem 2:

Find a formula for the general term of the sequence 3/16, 4/25, 5/36, 6/49..., assuming the pattern of the first few terms continues.

#### Problem 3:

What is the formula for the nth term in an arithmetic sequence: a1 , a1 + d, a1 + 2d, a1 + 3d, ...?

#### Problem 4:

Consider the geometric sequence 3, 6,12, 24, 48, ... Find a formula for an.

#### Problem 5:

A natural number (1,2,3,4,... ) is called a prime number if it is greater than 1 and cannot be written as the product of two smaller natural numbers.

What are the missing numbers in the following sequence of prime numbers?

2, 3, 5, 7, 11, _____, 17, 19, 23, 29, 31, _____, 41, 43, 47, 53, 59, 61, _____, 71, 73, 79, _____, 89, 97,...

#### Answer 1:

1/2n

#### Answer 2:

(n+2)/(n+3)2

#### Answer 3:

In an arithmetic sequence with steps of size d, an = a1+(n + 1) d

#### Answer 4:

The ratio between successive terms is r. The nth term in the sequence is:

an = rn-1 a1 = 2n-1 ( 3 ).

#### Answer 5:

13, 37, 67, 83

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