10 Questions
- Sequences are often represented using variables like a_n, and the entire sequence can be referred to as (a_n)_{n\in\N}.
notation
- Sequences can be manipulated similarly to functions, including shifting, adding, and finding the rate of change.
manipulations
- Recursive definitions can be used to define sequences, although they may be more difficult to work with than ______ formulas.
closed
- Comparing a given sequence to a known sequence with a closed formula can be a useful method for ______ the closed formula.
finding
- The sequence of partial sums can be formed by adding up the terms of a sequence up to a certain point and may have a closed formula that can be found from the closed formula of the original ______.
sequence
- Notation like (\d\sum_{k=1}^n a_k) is used to represent the ______ of the terms in a sequence from 1 to n.
sum
- Examples of known sequences include arithmetic and geometric sequences, which have closed formulas that can be used to find the closed formulas of other ______.
sequences
To find a closed formula for an arithmetic sequence, we need to start with the first term and then add the ______ a bunch of times.
common difference
A sequence with a constant ratio between successive terms is called a ______ sequence.
geometric
If a sequence starts with a term other than a_0, we can find the term that would have been a_0 by using the formula a_0 = ______.
(the given term) - (common difference or common ratio)*(index of the given term - 1)
Study Notes
Describing Sequences and Finding Closed Formulas
- A sequence is an ordered list of numbers, where the order of the numbers matters.
- Sequences are often represented using variables like a_n, and the entire sequence can be referred to as (a_n)_{n\in\N}.
- Sequences are a type of function, with the domain of natural numbers and a_n as their image.
- Sequences can be manipulated similarly to functions, including shifting, adding, and finding the rate of change.
- Recursive definitions can be used to define sequences, although they may be more difficult to work with than closed formulas.
- Finding closed formulas for sequences can be challenging and there is no one method for doing so.
- Comparing a given sequence to a known sequence with a closed formula can be a useful method for finding the closed formula.
- The sequence of partial sums can be formed by adding up the terms of a sequence up to a certain point and may have a closed formula that can be found from the closed formula of the original sequence.
- Notation like (\d\sum_{k=1}^n a_k) is used to represent the sum of the terms in a sequence from 1 to n.
- Sequences can also be multiplied using notation like (\d\prod_{k=1}^n a_k).
- Recursive definitions can be used to find the total number of push-ups done by Sam after the nth day of her push-up challenge.
- Examples of known sequences include arithmetic and geometric sequences, which have closed formulas that can be used to find the closed formulas of other sequences.
Test your knowledge of sequences and closed formulas with this quiz! From understanding the basics of sequences as an ordered list of numbers to manipulating them like functions, this quiz covers it all. You'll also be challenged to find closed formulas for sequences and compare them to known sequences. Plus, you'll get to practice using notation to represent sums and products of sequences. Whether you're a beginner or an expert, this quiz will put your knowledge to the test.
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