Write Explicit or Recursive Functions - Algebra
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Which phrase in context usually signals an arithmetic pattern: “adds a fixed amount” or “multiplies by a fixed factor”?
Which phrase in context usually signals an arithmetic pattern: “adds a fixed amount” or “multiplies by a fixed factor”?
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“Adds a fixed amount”. Constant addition indicates arithmetic/linear growth patterns.
“Adds a fixed amount”. Constant addition indicates arithmetic/linear growth patterns.
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What is the recursive formula for exponential growth starting at $P_0$ with factor $b$ per step?
What is the recursive formula for exponential growth starting at $P_0$ with factor $b$ per step?
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$P(0)=P_0$, and $P(n)=b\cdot P(n-1)$ for $n\ge 1$. Each step multiplies the previous value by the growth factor $b$.
$P(0)=P_0$, and $P(n)=b\cdot P(n-1)$ for $n\ge 1$. Each step multiplies the previous value by the growth factor $b$.
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What is the explicit formula if a taxi charges $4$ dollars plus $2$ dollars per mile for $m$ miles?
What is the explicit formula if a taxi charges $4$ dollars plus $2$ dollars per mile for $m$ miles?
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$C(m)=4+2m$. Fixed cost of $4$ plus variable cost of $2$ per mile.
$C(m)=4+2m$. Fixed cost of $4$ plus variable cost of $2$ per mile.
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What is the recursive formula for “start with $50$ and add $7$ each week,” written as $a_n$ with $a_1=50$?
What is the recursive formula for “start with $50$ and add $7$ each week,” written as $a_n$ with $a_1=50$?
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$a_1=50$, $a_n=a_{n-1}+7$. Base case and rule for adding 7 to each previous term.
$a_1=50$, $a_n=a_{n-1}+7$. Base case and rule for adding 7 to each previous term.
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What is the explicit formula for “start with $50$ and add $7$ each week,” written as $a_n$ with $a_1=50$?
What is the explicit formula for “start with $50$ and add $7$ each week,” written as $a_n$ with $a_1=50$?
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$a_n=50+7(n-1)$. Arithmetic sequence with $a_1=50$ and common difference 7.
$a_n=50+7(n-1)$. Arithmetic sequence with $a_1=50$ and common difference 7.
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What is the explicit formula if you start with $P_0$ and decrease by $p%$ each step for $n$ steps?
What is the explicit formula if you start with $P_0$ and decrease by $p%$ each step for $n$ steps?
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$P(n)=P_0\left(1-\frac{p}{100}\right)^n$. Convert percentage decrease to decimal and subtract from 1.
$P(n)=P_0\left(1-\frac{p}{100}\right)^n$. Convert percentage decrease to decimal and subtract from 1.
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What is the explicit formula for a value starting at $90$ and decreasing by $12%$ each year for $n$ years?
What is the explicit formula for a value starting at $90$ and decreasing by $12%$ each year for $n$ years?
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$V(n)=90\cdot (0.88)^n$. Exponential decay with 12% decrease means factor of 0.88.
$V(n)=90\cdot (0.88)^n$. Exponential decay with 12% decrease means factor of 0.88.
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What is the recursive formula for “start with $1000$ and increase by $8%$ each step,” using $P(0)$?
What is the recursive formula for “start with $1000$ and increase by $8%$ each step,” using $P(0)$?
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$P(0)=1000$, $P(n)=1.08\cdot P(n-1)$. Base case and rule for 8% growth each step.
$P(0)=1000$, $P(n)=1.08\cdot P(n-1)$. Base case and rule for 8% growth each step.
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What is the growth factor $b$ for an increase of $8%$ per step?
What is the growth factor $b$ for an increase of $8%$ per step?
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$b=1.08$. 8% increase means multiply by $1+0.08=1.08$.
$b=1.08$. 8% increase means multiply by $1+0.08=1.08$.
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What is the recursive formula for a value starting at $90$ and decreasing by $12%$ each year?
What is the recursive formula for a value starting at $90$ and decreasing by $12%$ each year?
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$V(0)=90$, $V(n)=0.88\cdot V(n-1)$. Base case and rule for multiplying by 0.88 each year.
$V(0)=90$, $V(n)=0.88\cdot V(n-1)$. Base case and rule for multiplying by 0.88 each year.
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Which phrase in context usually signals a geometric pattern: “adds a fixed amount” or “multiplies by a fixed factor”?
Which phrase in context usually signals a geometric pattern: “adds a fixed amount” or “multiplies by a fixed factor”?
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“Multiplies by a fixed factor”. Constant multiplication indicates geometric/exponential growth patterns.
“Multiplies by a fixed factor”. Constant multiplication indicates geometric/exponential growth patterns.
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What is the recursive formula for “start with $1000$ and decrease by $35%$ each step,” using $P(0)$?
What is the recursive formula for “start with $1000$ and decrease by $35%$ each step,” using $P(0)$?
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$P(0)=1000$, $P(n)=0.65\cdot P(n-1)$. Base case and rule for 35% decay each step.
$P(0)=1000$, $P(n)=0.65\cdot P(n-1)$. Base case and rule for 35% decay each step.
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What is the decay factor $b$ for a decrease of $35%$ per step?
What is the decay factor $b$ for a decrease of $35%$ per step?
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$b=0.65$. 35% decrease means multiply by $1-0.35=0.65$.
$b=0.65$. 35% decrease means multiply by $1-0.35=0.65$.
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What is the explicit formula if you start with $P_0$ and increase by $p%$ each step for $n$ steps?
What is the explicit formula if you start with $P_0$ and increase by $p%$ each step for $n$ steps?
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$P(n)=P_0\left(1+\frac{p}{100}\right)^n$. Convert percentage increase to decimal and add to 1.
$P(n)=P_0\left(1+\frac{p}{100}\right)^n$. Convert percentage increase to decimal and add to 1.
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Which model fits “population increases by $4%$ each year”: linear or exponential?
Which model fits “population increases by $4%$ each year”: linear or exponential?
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Exponential. Multiplying by a constant percent each year creates exponential growth.
Exponential. Multiplying by a constant percent each year creates exponential growth.
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Which model fits “population increases by $300$ each year”: linear or exponential?
Which model fits “population increases by $300$ each year”: linear or exponential?
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Linear. Adding a constant amount each year creates linear growth.
Linear. Adding a constant amount each year creates linear growth.
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What is the recursive formula for “start with $9$ and multiply by $-2$ each step,” using $a_1=9$?
What is the recursive formula for “start with $9$ and multiply by $-2$ each step,” using $a_1=9$?
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$a_1=9$, $a_n=-2\cdot a_{n-1}$. Base case and rule for multiplying by -2 each step.
$a_1=9$, $a_n=-2\cdot a_{n-1}$. Base case and rule for multiplying by -2 each step.
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What is the explicit formula for “start with $9$ and multiply by $-2$ each step,” using $a_1=9$?
What is the explicit formula for “start with $9$ and multiply by $-2$ each step,” using $a_1=9$?
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$a_n=9\cdot (-2)^{n-1}$. Geometric sequence with $a_1=9$ and common ratio -2.
$a_n=9\cdot (-2)^{n-1}$. Geometric sequence with $a_1=9$ and common ratio -2.
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What is the recursive formula for a geometric sequence with common ratio $r$?
What is the recursive formula for a geometric sequence with common ratio $r$?
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$a_1$ given, and $a_n=r\cdot a_{n-1}$ for $n\ge 2$. Each term multiplies the previous term by the common ratio.
$a_1$ given, and $a_n=r\cdot a_{n-1}$ for $n\ge 2$. Each term multiplies the previous term by the common ratio.
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What is the recursive formula for linear growth starting at $P_0$ with change $m$ per step?
What is the recursive formula for linear growth starting at $P_0$ with change $m$ per step?
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$P(0)=P_0$, and $P(n)=P(n-1)+m$ for $n\ge 1$. Each step adds the constant change $m$ to the previous value.
$P(0)=P_0$, and $P(n)=P(n-1)+m$ for $n\ge 1$. Each step adds the constant change $m$ to the previous value.
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What is the explicit formula for exponential growth starting at $P_0$ with factor $b$ per step?
What is the explicit formula for exponential growth starting at $P_0$ with factor $b$ per step?
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$P(n)=P_0\cdot b^n$. Exponential function with initial value and constant growth factor.
$P(n)=P_0\cdot b^n$. Exponential function with initial value and constant growth factor.
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What is the explicit formula for linear growth starting at $P_0$ with change $m$ per step ($n$ steps)?
What is the explicit formula for linear growth starting at $P_0$ with change $m$ per step ($n$ steps)?
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$P(n)=P_0+mn$. Linear function with initial value and constant rate of change.
$P(n)=P_0+mn$. Linear function with initial value and constant rate of change.
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What is the explicit formula for a geometric sequence with first term $a_1$ and common ratio $r$?
What is the explicit formula for a geometric sequence with first term $a_1$ and common ratio $r$?
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$a_n=a_1\cdot r^{n-1}$. Standard geometric sequence formula with first term and common ratio.
$a_n=a_1\cdot r^{n-1}$. Standard geometric sequence formula with first term and common ratio.
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What is the recursive formula for an arithmetic sequence with common difference $d$?
What is the recursive formula for an arithmetic sequence with common difference $d$?
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$a_1$ given, and $a_n=a_{n-1}+d$ for $n\ge 2$. Each term adds the common difference to the previous term.
$a_1$ given, and $a_n=a_{n-1}+d$ for $n\ge 2$. Each term adds the common difference to the previous term.
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What is the explicit formula for an arithmetic sequence with first term $a_1$ and common difference $d$?
What is the explicit formula for an arithmetic sequence with first term $a_1$ and common difference $d$?
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$a_n=a_1+(n-1)d$. Standard arithmetic sequence formula with first term and common difference.
$a_n=a_1+(n-1)d$. Standard arithmetic sequence formula with first term and common difference.
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What two parts must a recursive definition of $a_n$ always include?
What two parts must a recursive definition of $a_n$ always include?
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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.
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.
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What is the definition of a recursive formula for a sequence $a_n$?
What is the definition of a recursive formula for a sequence $a_n$?
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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.
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.
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What is the definition of an explicit formula for a sequence $a_n$?
What is the definition of an explicit formula for a sequence $a_n$?
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A rule that gives $a_n$ directly from $n$ (no prior terms needed). You can find any term directly using just the position $n$.
A rule that gives $a_n$ directly from $n$ (no prior terms needed). You can find any term directly using just the position $n$.
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What is the explicit expression for “start with $12$ and increase by $5$ each step” after $n$ steps?
What is the explicit expression for “start with $12$ and increase by $5$ each step” after $n$ steps?
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$P(n)=12+5n$. Linear function starting at 12 with slope 5.
$P(n)=12+5n$. Linear function starting at 12 with slope 5.
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What is the recursive rule for “start with $12$ and increase by $5$ each step” using $P(0)$?
What is the recursive rule for “start with $12$ and increase by $5$ each step” using $P(0)$?
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$P(0)=12$, $P(n)=P(n-1)+5$. Base case and rule for adding 5 each step.
$P(0)=12$, $P(n)=P(n-1)+5$. Base case and rule for adding 5 each step.
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