Module 4.2 Constructing Arithmetic Sequences
Transcript of Module 4.2 Constructing Arithmetic Sequences
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Module 4.2
Constructing Arithmetic Sequences
What is an arithmetic sequence?
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In an arithmetic sequence, the difference between consecutive terms is always equal. This difference, written as d, is called the common difference.
Consider this sequence, defined by the explicit rule π π = ππ +5
Domain
Range
The 2nd term minus the 1st term (9 β 7) is 2.The 3rd term minus the 2nd term (11 β 9) is 2.The 4th term minus the 3rd term (13 β 11) is 2. Etc.
So the common difference d is 2.
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In Module 4.1, we learned how to generate a sequence from an explicit rule.(By substituting every position number into the rule, one at a time.)
And we learned how to generate a sequence from a recursive rule.(By taking the previous term, and performing some operation on it.)
Now we want to do the opposite:
Given a sequence:1) Determine the explicit rule that was used to create it.2) Determine the recursive rule that was used to create it.
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1) Given a sequence, determine the explicit rule that was used to create it.
It can be done from: (a) a list of numbers(b) a table(c) a graph.
Use this formula: π π = π π + π π β π
The two variables you need are π π and π
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or π π = ____ π β π
or π π = βππ π β π + πππ
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This common difference d is $500, the amount deposited each month.Use this formula: π π = π π + π (π β π)The Explicit Rule is: π π = ππππ + πππ(π β π) or π π = πππ π β π + ππππ
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or π π = ππ π β π + ππ
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How can you tell from the graph that this is an arithmetic sequence? The points of the graph are in a straight line, indicating that there is a constant difference between consecutive terms.
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2) Given a sequence, determine the recursive rule that was used to create it.
It can be done from: (a) a list of numbers(b) a table.
Use this format: π π =?, π π = π π β π + π πππ π β₯ π
The two variables you need are π π and π
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This common difference d is $500, the amount deposited each month.And π π = πππ.Use this format: π π =?, π π = π π β π + π πππ π β₯ πRecursive Rule: π π = ππππ, π π = π π β π + πππ πππ π β₯ π
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Hint: The format for the explicit rule is π π = π π + π π β πIn our example, can you identify d and π(π)?
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