Introduction to Sequences Lesson

In this lesson, we will introduce the concept of a sequence in math. A sequence is a function that is used to compute an ordered list. Instead of the typical function notation f(x), we will see sequence notation a_n. The n is a reminder that the domain or set of allowable values for n is the set of natural numbers.
Natural Numbers: $${1, 2, 3, 4, 5, ...}$$ The general term of a sequence is given as a_n. If we want to find the first term of the sequence, we simply plug in a 1 for n and evaluate. Let's see this with an example.
Example #1: Find the first three terms of the sequence. $$a_n=11 - 20n$$ To find the first three terms, we start with the first natural number, which is 1. $$a_1=11 - 20(1)=-9$$ To find the second term, we plug in the next natural number, which is 2. $$a_2=11 - 20(2)=- 29$$ Lastly, to find the third term, we plug in the next natural number, which is 3. $$a_3=11 - 20(3)=-49$$ The first three terms of the sequence: $$-9, -29, -49$$ In some cases, our sequence is defined by a recursive definition. This means each term after the first term or the first few terms is defined as some expression that involves the previous term or terms. Let's see an example.
Example #2: Find the first three terms of the sequence. $$a_{n + 1}=a_n \cdot 6$$ $$a_1=1$$ Here, we are given the first term, which is 1. For the next term, we need to think about the notation. a_{n + 1} in this case, would be a₂, which means a_n would be one less or a₁. All this is saying is to plug in the previous term and evaluate. $$a_2=a_1 \cdot 6$$ $$a_2=1 \cdot 6=6$$ Our second term is 6. To find the third term, now we plug in the second term, which is 6, and apply the same process. $$a_3=a_2 \cdot 6$$ $$a_3=6 \cdot 6=36$$ Our third term is 36.
The first three terms of the sequence: $$1, 6, 36$$

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