50 lines
2.6 KiB
Markdown
50 lines
2.6 KiB
Markdown
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---
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title: Recursive Formulas for Arithmetic Sequences
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---
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## Recursive Formulas for Arithmetic Sequences
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<!-- The article goes here, in GitHub-flavored Markdown. Feel free to add YouTube videos, images, and CodePen/JSBin embeds -->
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### What is an Arithmetic Sequence?
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A **sequence** is list of numbers where the same operation(s) is done to one number in order to get the next. **Arithmetic sequences**
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specifically refer to sequences constructed by adding or subtracting a value-called the **common difference**- to get the next term. In
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order to efficiently talk about a sequence, we use a formula that builds the sequence when a list of indices are put in. Typically, these formulas are given one-letter names, followed by a parameter in parentheses, and the expression that builds the sequence on the right hand side.
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`a(n) = n + 1`
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Above is an example of a formula for an arithmetic sequence.
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### Examples
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Sequence | Formula
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---------|---------
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1, 2, 3, 4, ... | a(n) = n + 1
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3, 8, 13, 18, ... | b(n) = 5n - 2
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### A Recursive Formula
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Note: Mathematicians start counting at 1, so by convention, `n=1` is the first term. So we must define what the first term is. Then we have
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to figure out and include the common difference. Taking a look at the examples again,
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Sequence | Formula | Recursive Formula
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---------|---------|-------------------
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1, 2, 3, 4, ... | a(n) = n + 1 | a(n) = a(n-1) + 1, a(1) = 1
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3, 8, 13, 18, ... | b(n) = 5n - 2 | b(n) = b(n-1) + 5, b(1) = 3
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### Finding the Formula (given a sequence with the first term)
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1. Figure out the common difference
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Pick a term in the sequence and subtract the term that comes before it.
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2. Construct the formula
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The formula has the form: `a(n) = a(n-1) + [common difference], a(1) = [first term]`
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### Finding the Formula (given a sequence without the first term)
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1. Figure out the common difference
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Pick a term in the sequence and subtract the term that comes before it.
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2. Find the first term
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i. Pick a term in the sequence, call it `k` and call its index `h`
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ii. first term = k - (h-1)*(common difference)
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3. Construct the formula
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The formula has the form: `a(n) = a(n-1) + [common difference], a(1) = [first term]`
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#### More Information:
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<!-- Please add any articles you think might be helpful to read before writing the article -->
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For more information on this topic, visit
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- [Wikipedia](https://en.wikipedia.org/wiki/Arithmetic_progression)
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- [Khan Academy](https://www.khanacademy.org/math/algebra/sequences/constructing-arithmetic-sequences/a/writing-recursive-formulas-for-arithmetic-sequences)
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