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Questions and Answers
What is an ordered list of numbers called?
sequence
What notation is used to designate the nth term of a sequence?
subscript notation an
What are the differences in the first row of a difference table called?
first differences
In a sequence where first differences are the same, how is the next term predicted?
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What are the second differences in a difference table?
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How was the next term predicted in the example sequence 2, 7, 24, 59, 118, 207?
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What is the nth-term formula discovered in the tile sequence example?
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How many tiles are in the eighth figure of the sequence?
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Which figure will consist of exactly 320 tiles?
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Study Notes
Terms of a Sequence
- A sequence is an ordered list of numbers, such as 5, 14, 27, 44, 65, denoted by terms separated by commas.
- Each term has a designated position; for example, 5 is the first term (a1), 14 is the second (a2), and so on.
- The ellipsis (...) indicates the sequence continues beyond the last number listed.
Difference Tables
- A difference table helps identify patterns in sequences by showing the differences between successive terms.
- First differences are calculated by subtracting each term from the subsequent term.
- If first differences are constant (the same value), the sequence can be described as arithmetic, where each term increases by a fixed amount.
Predicting Terms
- If first differences are not constant, compute successive differences (second differences) to look for patterns.
- Second and third differences may help identify polynomial relationships within the sequence.
- The next term can often be predicted by continuing the observed pattern in the differences.
Example of Predicting a Term
- Use the sequence 2, 7, 24, 59, 118, 207, creating a difference table to observe changes.
- In this case, the third differences were constant, allowing the prediction that the next term would be 332.
nth-Term Formula for a Sequence
- An nth-term formula generates terms of a sequence based on a defined pattern.
- Patterns may appear visually, such as with tiles arranged in figures, leading to specific formulas.
Finding an nth-Term Formula
- Example discussion involved determining tile patterns, finding the number of tiles in a figure based on its position (n).
- The general formula derived from the pattern was an = 3n - 1.
- This formula can be used to calculate how many tiles are in a specific figure, like the eighth figure (23 tiles) based on substituting n=8.
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Description
This quiz explores the concepts of sequences, including their terms and how to analyze them using difference tables. You will learn to identify patterns in sequences and predict future terms based on their differences. Get ready to test your understanding of arithmetic and polynomial sequences!
