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Questions and Answers
- 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 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.
- 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.
- 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.
- Comparing a given sequence to a known sequence with a closed formula can be a useful method for ______ the closed formula.
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- 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 ______.
- 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 ______.
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- Notation like (\d\sum_{k=1}^n a_k) is used to represent the ______ of the terms in a sequence from 1 to n.
- Notation like (\d\sum_{k=1}^n a_k) is used to represent the ______ of the terms in a sequence from 1 to n.
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- Examples of known sequences include arithmetic and geometric sequences, which have closed formulas that can be used to find the closed formulas of other ______.
- Examples of known sequences include arithmetic and geometric sequences, which have closed formulas that can be used to find the closed formulas of other ______.
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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.
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.
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A sequence with a constant ratio between successive terms is called a ______ sequence.
A sequence with a constant ratio between successive terms is called a ______ sequence.
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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 = ______.
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 = ______.
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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.
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