Algebra Flashcards: Write Explicit Or Recursive Functions

What this deck covers

This deck focuses on Write Explicit Or Recursive Functions, giving you a quick way to review the definitions, rules, and examples that matter most for Algebra.

How to use these flashcards

Work through these flashcards in short sessions. Try to answer each prompt before flipping the card, then revisit any cards you miss until the explanation feels automatic.

Flashcard 1: What does the exponent $n-1$ represent in $a_n=a_1\cdot r^{n-1}$?

Answer: The number of times the ratio $r$ is applied from term $1$ to term $n$. Shows how many times we multiply by $r$ to get from $a_1$ to $a_n$.

Flashcard 2: What is the recursive formula for "start with $9$ and multiply by $-2$ each step," using $a_1=9$?

Answer: $a_1=9$, $a_n=-2\cdot a_{n-1}$. Base case and rule for multiplying by -2 each step.

Flashcard 3: What is the recursive formula for "start with $1000$ and increase by $8\%$ each step," using $P(0)$?

Answer: $P(0)=1000$, $P(n)=1.08\cdot P(n-1)$. Base case and rule for 8% growth each step.

Flashcard 4: What is the explicit formula for a geometric sequence with first term $a_1$ and common ratio $r$?

Answer: $a_n=a_1\cdot r^{n-1}$. Standard geometric sequence formula with first term and common ratio.

Flashcard 5: What is the explicit formula if you start with $P_0$ and decrease by $p\%$ each step for $n$ steps?

Answer: $P(n)=P_0\left(1-\frac{p}{100}\right)^n$. Convert percentage decrease to decimal and subtract from 1.

Flashcard 6: What is the recursive formula for a geometric sequence with common ratio $r$?

Answer: $a_1$ given, and $a_n=r\cdot a_{n-1}$ for $n\ge 2$. Each term multiplies the previous term by the common ratio.

Flashcard 7: What is the explicit formula for an arithmetic sequence with first term $a_1$ and common difference $d$?

Answer: $a_n=a_1+(n-1)d$. Standard arithmetic sequence formula with first term and common difference.

Flashcard 8: What is the common ratio $r$ for the geometric sequence $2,6,18,54,\dots$?

Answer: $r=3$. Each term is tripled: $6÷2=3$, $18÷6=3$.

Flashcard 9: What is the explicit formula for an arithmetic sequence with $a_1=7$ and $d=4$?

Answer: $a_n=7+4(n-1)$. Arithmetic sequence with first term 7 and common difference 4.

Flashcard 10: What is the explicit expression for "start with $200$ and triple each hour" after $n$ hours?

Answer: $P(n)=200\cdot 3^n$. Exponential function with initial value 200 and growth factor 3.

Flashcard 11: What is the explicit formula for the arithmetic sequence $3,8,13,18,\dots$?

Answer: $a_n=3+5(n-1)$. Arithmetic sequence formula with $a_1=3$ and $d=5$.

Flashcard 12: What is the explicit expression for "start with $12$ and increase by $5$ each step" after $n$ steps?

Answer: $P(n)=12+5n$. Linear function starting at 12 with slope 5.

Flashcard 13: Which model fits "population increases by $300$ each year": linear or exponential?

Answer: Linear. Adding a constant amount each year creates linear growth.

Flashcard 14: What is the recursive rule for "start with $500$ and keep $90\%$ each month" using $P(0)$?

Answer: $P(0)=500$, $P(n)=0.9\cdot P(n-1)$. Base case and rule for keeping 90% each step.

Flashcard 15: What is the recursive formula for a value starting at $90$ and decreasing by $12\%$ each year?

Answer: $V(0)=90$, $V(n)=0.88\cdot V(n-1)$. Base case and rule for multiplying by 0.88 each year.

Flashcard 16: What is the definition of an explicit formula for a sequence $a_n$?

Answer: A rule that gives $a_n$ directly from $n$ (no prior terms needed). You can find any term directly using just the position $n$.

Flashcard 17: What is the explicit expression for "start with $80$ and decrease by $3$ each day" after $n$ days?

Answer: $P(n)=80-3n$. Linear function starting at 80 with negative slope -3.

Flashcard 18: What is the explicit formula for "start with $9$ and multiply by $-2$ each step," using $a_1=9$?

Answer: $a_n=9\cdot (-2)^{n-1}$. Geometric sequence with $a_1=9$ and common ratio -2.

Flashcard 19: What is the recursive formula for a geometric sequence with $a_1=6$ and $r=2$?

Answer: $a_1=6$, $a_n=2\cdot a_{n-1}$. Base case and rule for doubling each previous term.

Flashcard 20: What is the explicit formula for the geometric sequence $2,6,18,54,\dots$?

Answer: $a_n=2\cdot 3^{n-1}$. Geometric sequence formula with $a_1=2$ and $r=3$.

Flashcard 21: What is the recursive rule for "start with $200$ and triple each hour" using $P(0)$?

Answer: $P(0)=200$, $P(n)=3\cdot P(n-1)$. Base case and rule for tripling each step.

Flashcard 22: Which phrase in context usually signals an arithmetic pattern: "adds a fixed amount" or "multiplies by a fixed factor"?

Answer: "Adds a fixed amount". Constant addition indicates arithmetic/linear growth patterns.

Flashcard 23: What is the growth factor $b$ for an increase of $8\%$ per step?

Answer: $b=1.08$. 8% increase means multiply by $1+0.08=1.08$.

Flashcard 24: What is the explicit formula if a taxi charges $4$ dollars plus $2$ dollars per mile for $m$ miles?

Answer: $C(m)=4+2m$. Fixed cost of $4$ plus variable cost of $2$ per mile.

Flashcard 25: What is the common difference $d$ for the arithmetic sequence $3,8,13,18,\dots$?

Answer: $d=5$. Each term increases by 5: $8-3=5$, $13-8=5$.

Flashcard 26: What is the explicit formula for "start with $50$ and add $7$ each week," written as $a_n$ with $a_1=50$?

Answer: $a_n=50+7(n-1)$. Arithmetic sequence with $a_1=50$ and common difference 7.

Flashcard 27: What is the explicit formula for a value starting at $90$ and decreasing by $12\%$ each year for $n$ years?

Answer: $V(n)=90\cdot (0.88)^n$. Exponential decay with 12% decrease means factor of 0.88.

Flashcard 28: What is the explicit formula for a geometric sequence with $a_1=6$ and $r=2$?

Answer: $a_n=6\cdot 2^{n-1}$. Geometric sequence with first term 6 and common ratio 2.

Flashcard 29: What is the explicit formula for linear growth starting at $P_0$ with change $m$ per step ($n$ steps)?

Answer: $P(n)=P_0+mn$. Linear function with initial value and constant rate of change.

Flashcard 30: Which model fits "population increases by $4\%$ each year": linear or exponential?

Answer: Exponential. Multiplying by a constant percent each year creates exponential growth.

Flashcard 31: What is the common difference $d$ if consecutive terms are $a_{n-1}$ and $a_n$ in an arithmetic sequence?

Answer: $d=a_n-a_{n-1}$. The common difference is the constant amount added each step.

Flashcard 32: What is the recursive formula for an arithmetic sequence with $a_1=7$ and $d=4$?

Answer: $a_1=7$, $a_n=a_{n-1}+4$. Base case and rule for adding 4 to each previous term.

Flashcard 33: What is the common ratio $r$ if consecutive terms are $a_{n-1}$ and $a_n$ in a geometric sequence?

Answer: $r=\frac{a_n}{a_{n-1}}$. The common ratio is the constant factor multiplied each step.

Flashcard 34: What is the explicit formula for a geometric sequence with $a_1=10$ and $r=\frac{1}{2}$?

Answer: $a_n=10\left(\frac{1}{2}\right)^{n-1}$. Geometric sequence with first term 10 and common ratio 1/2.

Flashcard 35: What is the explicit formula for an arithmetic sequence with $a_1=-3$ and $d=-2$?

Answer: $a_n=-3-2(n-1)$. Arithmetic sequence with first term -3 and common difference -2.

Flashcard 36: What is the definition of a recursive formula for a sequence $a_n$?

Answer: A rule that gives $a_n$ using earlier term(s), plus a starting value. Each term depends on the previous term(s), so you build sequentially.

Flashcard 37: What is the explicit formula for earnings of $15$ dollars per hour for $h$ hours plus a $20$ dollar bonus?

Answer: $E(h)=15h+20$. Variable earnings of $15 per hour plus fixed bonus of $20.

Flashcard 38: What is the recursive formula for an arithmetic sequence with common difference $d$?

Answer: $a_1$ given, and $a_n=a_{n-1}+d$ for $n\ge 2$. Each term adds the common difference to the previous term.

Flashcard 39: What is the explicit formula if you start with $P_0$ and increase by $p\%$ each step for $n$ steps?

Answer: $P(n)=P_0\left(1+\frac{p}{100}\right)^n$. Convert percentage increase to decimal and add to 1.

Flashcard 40: What two parts must a recursive definition of $a_n$ always include?

Answer: A starting term (like $a_1$) and a recursion rule (like $a_n=a_{n-1}+d$). You need a base case and a pattern to find the next term.

Flashcard 41: What does $n-1$ represent in the arithmetic explicit formula $a_n=a_1+(n-1)d$?

Answer: The number of steps from term $1$ to term $n$. Shows how many times we add $d$ to get from $a_1$ to $a_n$.

Flashcard 42: What is the decay factor $b$ for a decrease of $35\%$ per step?

Answer: $b=0.65$. 35% decrease means multiply by $1-0.35=0.65$.

Flashcard 43: What is the recursive formula for "start with $50$ and add $7$ each week," written as $a_n$ with $a_1=50$?

Answer: $a_1=50$, $a_n=a_{n-1}+7$. Base case and rule for adding 7 to each previous term.

Flashcard 44: What is the explicit formula for a balance starting at $120$ and increasing by $6\%$ each month for $n$ months?

Answer: $B(n)=120\cdot (1.06)^n$. Exponential growth with 6% increase means factor of 1.06.

Flashcard 45: What is the explicit expression for "start with $500$ and keep $90\%$ each month" after $n$ months?

Answer: $P(n)=500\cdot (0.9)^n$. Exponential decay with initial value 500 and decay factor 0.9.

Flashcard 46: What is the recursive formula for linear growth starting at $P_0$ with change $m$ per step?

Answer: $P(0)=P_0$, and $P(n)=P(n-1)+m$ for $n\ge 1$. Each step adds the constant change $m$ to the previous value.

Flashcard 47: What is the explicit formula for exponential growth starting at $P_0$ with factor $b$ per step?

Answer: $P(n)=P_0\cdot b^n$. Exponential function with initial value and constant growth factor.

Flashcard 48: What is the recursive rule for "start with $12$ and increase by $5$ each step" using $P(0)$?

Answer: $P(0)=12$, $P(n)=P(n-1)+5$. Base case and rule for adding 5 each step.

Flashcard 49: What is the recursive formula for exponential growth starting at $P_0$ with factor $b$ per step?

Answer: $P(0)=P_0$, and $P(n)=b\cdot P(n-1)$ for $n\ge 1$. Each step multiplies the previous value by the growth factor $b$.

Flashcard 50: What is the recursive rule for "start with $80$ and decrease by $3$ each day" using $P(0)$?

Answer: $P(0)=80$, $P(n)=P(n-1)-3$. Base case and rule for subtracting 3 each step.

Flashcard 51: What is the recursive formula for "start with $1000$ and decrease by $35\%$ each step," using $P(0)$?

Answer: $P(0)=1000$, $P(n)=0.65\cdot P(n-1)$. Base case and rule for 35% decay each step.

Flashcard 52: Which phrase in context usually signals a geometric pattern: "adds a fixed amount" or "multiplies by a fixed factor"?

Answer: "Multiplies by a fixed factor". Constant multiplication indicates geometric/exponential growth patterns.

Flashcard 53: What is the recursive formula for the sequence $2,6,18,54,\dots$ using $a_1$?

Answer: $a_1=2$, $a_n=3\cdot a_{n-1}$. Base case and rule for tripling each previous term.

Flashcard 54: What is the recursive formula for a balance starting at $120$ and increasing by $6\%$ each month?

Answer: $B(0)=120$, $B(n)=1.06\cdot B(n-1)$. Base case and rule for multiplying by 1.06 each month.