6.7 KiB
| title | TARGET DECK | FILE TAGS | tags | ||
|---|---|---|---|---|---|
| Geometric Sequence | Obsidian::STEM | algebra::sequence |
|
Overview
A geometric sequence (a_n)_{n \geq 0}^r is a sequence in which each term differs by a constant multiple of r. Given initial term a, it has recursive definition $a_n = ra_{n-1} \text{ with } a_0 = aand closed formulaa_n = ar^n.$
%%ANKI Basic What makes a sequence a geometric sequence? Back: Each term is the same constant multiple from the previous. Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.
END%%
%%ANKI Basic What two properties are necessary to characterize a geometric sequence? Back: The initial term and the common ratio. Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.
END%%
%%ANKI
Basic
What does it mean for the common ratio of a geometric sequence (a_n) to be r?
Back: a_n = a_{n-1} \cdot r for all valid values of n.
Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.
END%%
%%ANKI
Basic
Let (a_n) be a geometric sequence. What term refers to the ratio between terms?
Back: The common ratio.
Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.
END%%
%%ANKI Basic Why is the "common ratio" of a geometric sequence named the way it is? Back: It is shared between all successive terms. Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.
END%%
%%ANKI Basic The term "common ratio" is related to what kind of sequence? Back: A geometric sequence. Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.
END%%
%%ANKI
Basic
Let (a_n)_{n \geq 0}^r be a geometric sequence. What is its recursive definition's recurrence relation?
Back: a_n = a_{n-1} \cdot r
Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.
END%%
%%ANKI
Basic
Let (a_n)_{n \geq 0}^r be a geometric sequence. What is the $n$th term's closed formula?
Back: a_n = a_0 \cdot r^n
Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.
END%%
%%ANKI Cloze {1:Arithmetic} sequences are characterized by the common {2:difference}. {2:Geometric} sequences are characterized by the common {1:ratio}. Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.
END%%
%%ANKI
Basic
Let (a_n)_{n \geq 0}^r be a geometric sequence. What is the closed formula of \sum_{k=0}^n a_k?
Back: \frac{a_0(1 - r^{n+1})}{1 - r}
Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.
END%%
%%ANKI
Basic
Let (a_n)_{n \geq 1}^r be a geometric sequence. What is the closed formula of \sum_{k=1}^n a_k?
Back: \frac{a_1(1 - r^n)}{1 - r}
Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.
END%%
%%ANKI
Basic
Let (a_n)_{n \geq 1}^r be a geometric sequence. What does term n correspond to in the following? \sum a_k = \frac{a_1(1 - r^n)}{1 - r}
Back: The number of terms in the summation.
Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.
END%%
%%ANKI
Basic
Let (a_n)_{n \geq 1}^r be a geometric sequence. How do you expand \sum_{k=1}^n a_k to derive its closed formula?
Back:
\begin{matrix}
S & = & a_1r^0 & + & a_1r^1 & + & \cdots & + & a_1r^{n-1} & + & 0 \\
rS & = & 0 & + & a_1r^1 & + & \cdots & + & a_1r^{n-1} & + & a_1r^n \\
\hline
S - rS & = & a_1r^0 & & & & & & & - & a_1r^n
\end{matrix}$$
Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
<!--ID: 1709666305441-->
END%%
%%ANKI
Basic
Let $(a_n)_{n \geq 1}^r$ be a geometric sequence. How is term $1 - r$ derived in the following? $$\sum a_k = \frac{a_1(1 - r^n)}{1 - r}$$
Back: Given $S = \sum a_k$, by factoring out $S$ from $S - rS$.
Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
<!--ID: 1709666356524-->
END%%
%%ANKI
Cloze
You can find the partial sums of {geometric} sequences using the "multiply and subtract" strategy.
Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
<!--ID: 1709666305444-->
END%%
%%ANKI
Cloze
{1:Reverse} and {1:add} arithmetic sequences. {2:Multiply} and {2:subtract} geometric sequences.
Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
<!--ID: 1709666305447-->
END%%
## Bibliography
* Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).