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

- Screen 2 defines
*arithmetic sequence*and*common difference*, and then screen 3 has some questions: (1) What is the common difference, (2) which is an arithmetic sequence? - Screen 4 introduces
*terms*. First term, second term, $n$th term, etc. - Screen 5 introduces $a_n$ notation, starting with the
*recursive*formula $a_n = a_{n-1} + d$, with (screen 6) $a_1$ as the first term. - Screen 7 has a student write the recursive formula for a given arithmetic sequence.
- Screen 10 introduces the
*explicit*formula, $a_n = a_1 + \left(n-1\right)d$. - Screen 11 has a student find the $n$th term of a sequence, given the explicit formula.
- Screen 13 introduces some word problems, and 14.
- Screen 15 discusses converting between the two forms of formulas here (recursive and explicit).

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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