Unit 7: Sequences and Functions

7.1 Arithmetic Sequences

Let's start with notes taken throughout the lesson itself:

Arithmetic Sequences

Consider the sequence of numbers $3, 7, 11, 15, 19, \cdots$.

The first term of that sequence is $a_1 = 3$.

The second term of that sequence is $a_2 = 7$. We also know that $a_3=11$, $a_4=15$, and $a_5=19$.

From each term to the next we are adding $4$. This means that the common difference of this sequence is $d=4$. We can find it by subtracting any two terms: $19-15=4$, $15-11=4$, $11-7=4$, and $7-3=4$.


Consider the sequence of numbers $2, 6, 18, 54, \cdots$.

This sequence does not have a common difference, because $6-2=4$ but $18-6=12$. This is not an arithmetic sequence.

Recursive Form of an Arithmetic Sequence

The recursive form of an arithmetic sequence says where the sequence starts, and what happens from one term to the next.

Consider the sequence of numbers $3, 7, 11, 15, 19, \cdots$.

Where does the sequence start? $a_1=3$.

What happens as we move from one term to the next? We add $d=4$. $a_n=a_{n-1}+4$.

$\begin{aligned}a_1&=3\\a_n&=a_{n-1}+4\end{aligned}$

Explicit Form of an Arithmetic Sequence

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Converting between Recursive and Explicit Forms of an Arithmetic Sequence

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Converting from Recursive to Explicit Forms of an Arithmetic Sequence

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Converting from Explicit to Recursive Forms of an Arithmetic Sequence

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