Card 1 of 20
EasyStudy card 1
Prompt
What two parts does a recursive definition require?
Answer
An initial condition and a recurrence rule.
MathChapter 12: Composition, Recursion, and Exponential Functions
Flashcards
Recursive Formula Flashcards
Twenty concise cards on initial conditions, recurrence rules, indexing, iteration, and real-world recursive models.
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Card 1 of 20. Answer hidden.
Card 2 of 20
EasyStudy card 2
Prompt
What does an initial condition do?
Answer
It states the starting term and its index.
Card 3 of 20
EasyStudy card 3
Prompt
What does a recurrence rule do?
Answer
It defines a new term from one or more preceding terms.
Card 4 of 20
EasyStudy card 4
Prompt
What term follows \(a_0\)?
Answer
\(a_1\).
Card 5 of 20
EasyStudy card 5
Prompt
What term follows \(a_1\)?
Answer
\(a_2\).
Card 6 of 20
EasyStudy card 6
Prompt
In \(a_n=a_{n-1}+4\), which value is used to find \(a_n\)?
Answer
The immediately preceding term \(a_{n-1}\).
Card 7 of 20
EasyStudy card 7
Prompt
Should the initial term be reused at every recursive step?
Answer
No. Each newly calculated term becomes the next previous term.
Card 8 of 20
MediumStudy card 8
Prompt
Why calculate intermediate terms explicitly?
Answer
They prevent skipped-state and index-shift errors.
Card 9 of 20
MediumStudy card 9
Prompt
What does \(a_n=2a_{n-1}\) do each step?
Answer
It doubles the previous term.
Card 10 of 20
MediumStudy card 10
Prompt
What does \(a_n=a_{n-1}+n\) require besides the previous value?
Answer
The current index \(n\).
Card 11 of 20
MediumStudy card 11
Prompt
How is repeated self-composition recursive?
Answer
Each function output becomes the input for the next application of the same function.
Card 12 of 20
MediumStudy card 12
Prompt
What does \(f(f(f(x)))\) require?
Answer
Three sequential applications of \(f\), recording each intermediate output.
Card 13 of 20
MediumStudy card 13
Prompt
In \(A_n=1.02A_{n-1}+50\), which operation happens first?
Answer
Multiply the previous amount by \(1.02\), then add \(50\).
Card 14 of 20
MediumStudy card 14
Prompt
What would \(1.02(A_{n-1}+50)\) mean instead?
Answer
Add \(50\) first, then multiply the entire sum; it is a different recurrence.
Card 15 of 20
MediumStudy card 15
Prompt
Why delay rounding in a decimal recurrence?
Answer
Rounded intermediate states can accumulate error in later terms.
Card 16 of 20
MediumStudy card 16
Prompt
Recursive versus explicit formula: what is the distinction?
Answer
A recursive rule depends on earlier terms; an explicit rule computes a term directly from its index.
Card 17 of 20
HardStudy card 17
Prompt
If \(a_0=2\) and \(a_n=3a_{n-1}+1\), find \(a_2\).
Answer
\(a_1=7\), then \(a_2=22\).
Card 18 of 20
HardStudy card 18
Prompt
If \(b_1=10\) and \(b_n=b_{n-1}-2n\), find \(b_3\).
Answer
\(b_2=6\), then \(b_3=0\).
Card 19 of 20
HardStudy card 19
Prompt
What three columns expose most recurrence mistakes?
Answer
Index, previous term, and substituted rule; a fourth column can record the new term.
Card 20 of 20
HardStudy card 20
Prompt
How do you validate a proposed recurrence from a list?
Answer
Check its initial index and test at least two consecutive transitions.
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