# Special Sequences and How They Are Generated

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• 0:02 Special Sequences
• 0:43 Triangular and…
• 2:19 Square and Cube Sequences
• 3:06 Fibonacci Sequence
• 4:08 Lesson Summary
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Instructor: Yuanxin (Amy) Yang Alcocer

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

Sequences are interesting things in math and in nature. The most interesting thing is that these sequences have a pattern to how they are generated. We will learn about a few of them in this video lesson.

## 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.

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