Arithmetic and Geometric Sequences as Functions - Algebra 2
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What is the formula for the common difference $d$ using two terms $a_k$ and $a_m$ of an arithmetic sequence?
What is the formula for the common difference $d$ using two terms $a_k$ and $a_m$ of an arithmetic sequence?
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$d=\frac{a_m-a_k}{m-k}$. Difference formula: change in terms divided by change in positions.
$d=\frac{a_m-a_k}{m-k}$. Difference formula: change in terms divided by change in positions.
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Find $a_7$ for the arithmetic sequence $a_n=2+5(n-1)$.
Find $a_7$ for the arithmetic sequence $a_n=2+5(n-1)$.
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$a_7=32$. Using $a_n=2+5(n-1)$: $a_7=2+5(6)=2+30=32$.
$a_7=32$. Using $a_n=2+5(n-1)$: $a_7=2+5(6)=2+30=32$.
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Find $a_1$ for a geometric sequence with $a_3=36$ and $r=3$.
Find $a_1$ for a geometric sequence with $a_3=36$ and $r=3$.
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$a_1=4$. Working backwards: $a_1=\frac{a_3}{r^2}=\frac{36}{3^2}=\frac{36}{9}=4$.
$a_1=4$. Working backwards: $a_1=\frac{a_3}{r^2}=\frac{36}{3^2}=\frac{36}{9}=4$.
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Find $a_1$ for a geometric sequence with $a_3=36$ and $r=3$.
Find $a_1$ for a geometric sequence with $a_3=36$ and $r=3$.
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$a_1=4$. Working backwards: $a_1=\frac{a_3}{r^2}=\frac{36}{3^2}=\frac{36}{9}=4$.
$a_1=4$. Working backwards: $a_1=\frac{a_3}{r^2}=\frac{36}{3^2}=\frac{36}{9}=4$.
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Translate to explicit form: $a_1=7$, $a_n=a_{n-1}-2$ for $n\ge^2$.
Translate to explicit form: $a_1=7$, $a_n=a_{n-1}-2$ for $n\ge^2$.
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$a_n=7+(n-1)(-2)$. Converting recursive to explicit: $d=-2$, so $a_n=7+(n-1)(-2)$.
$a_n=7+(n-1)(-2)$. Converting recursive to explicit: $d=-2$, so $a_n=7+(n-1)(-2)$.
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Translate to explicit form: $a_1=5$, $a_n=\frac{1}{2}a_{n-1}$ for $n\ge^2$.
Translate to explicit form: $a_1=5$, $a_n=\frac{1}{2}a_{n-1}$ for $n\ge^2$.
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$a_n=5\left(\frac{1}{2}\right)^{n-1}$. Converting recursive to explicit: $r=\frac{1}{2}$, so $a_n=5\left(\frac{1}{2}\right)^{n-1}$.
$a_n=5\left(\frac{1}{2}\right)^{n-1}$. Converting recursive to explicit: $r=\frac{1}{2}$, so $a_n=5\left(\frac{1}{2}\right)^{n-1}$.
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What is the explicit formula for the arithmetic sequence with $a_1=12$ and $d=-3$?
What is the explicit formula for the arithmetic sequence with $a_1=12$ and $d=-3$?
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$a_n=12+(n-1)(-3)$. Uses the standard explicit formula with $a_1=12$ and $d=-3$.
$a_n=12+(n-1)(-3)$. Uses the standard explicit formula with $a_1=12$ and $d=-3$.
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What is the explicit formula for the geometric sequence with $a_1=2$ and $r=\frac{1}{4}$?
What is the explicit formula for the geometric sequence with $a_1=2$ and $r=\frac{1}{4}$?
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$a_n=2\left(\frac{1}{4}\right)^{n-1}$. Uses the standard explicit formula with $a_1=2$ and $r=\frac{1}{4}$.
$a_n=2\left(\frac{1}{4}\right)^{n-1}$. Uses the standard explicit formula with $a_1=2$ and $r=\frac{1}{4}$.
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Identify the arithmetic sequence: $3,7,11,15,\dots$; what is the common difference $d$?
Identify the arithmetic sequence: $3,7,11,15,\dots$; what is the common difference $d$?
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$d=4$. Each term increases by $4$: $7-3=4$, $11-7=4$, etc.
$d=4$. Each term increases by $4$: $7-3=4$, $11-7=4$, etc.
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What is the recursive definition for the arithmetic sequence with $a_1=-5$ and $d=6$?
What is the recursive definition for the arithmetic sequence with $a_1=-5$ and $d=6$?
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$a_1=-5$; $a_n=a_{n-1}+6$. Standard recursive form with given first term and difference.
$a_1=-5$; $a_n=a_{n-1}+6$. Standard recursive form with given first term and difference.
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Translate to recursive form: $a_n=3+4(n-1)$ with $n\ge^1$.
Translate to recursive form: $a_n=3+4(n-1)$ with $n\ge^1$.
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$a_1=3$; $a_n=a_{n-1}+4$. Converting explicit to recursive: $a_1=3$ and $d=4$.
$a_1=3$; $a_n=a_{n-1}+4$. Converting explicit to recursive: $a_1=3$ and $d=4$.
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Translate to recursive form: $a_n=8\cdot(-3)^{n-1}$ with $n\ge^1$.
Translate to recursive form: $a_n=8\cdot(-3)^{n-1}$ with $n\ge^1$.
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$a_1=8$; $a_n=-3,a_{n-1}$. Converting explicit to recursive: $a_1=8$ and $r=-3$.
$a_1=8$; $a_n=-3,a_{n-1}$. Converting explicit to recursive: $a_1=8$ and $r=-3$.
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Find and correct the error: Geometric explicit written as $a_n=a_1\cdot r^{n}$.
Find and correct the error: Geometric explicit written as $a_n=a_1\cdot r^{n}$.
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Correct: $a_n=a_1\cdot r^{n-1}$. The exponent should be $(n-1)$, not $n$, for standard form.
Correct: $a_n=a_1\cdot r^{n-1}$. The exponent should be $(n-1)$, not $n$, for standard form.
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Which option is the common difference for the arithmetic sequence $-2,1,4,7,\dots$?
Which option is the common difference for the arithmetic sequence $-2,1,4,7,\dots$?
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$d=3$. Each term increases by the same amount: $1-(-2)=3$.
$d=3$. Each term increases by the same amount: $1-(-2)=3$.
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Which option is the common ratio for the geometric sequence $-4,12,-36,108,\dots$?
Which option is the common ratio for the geometric sequence $-4,12,-36,108,\dots$?
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$r=-3$. Each term is multiplied by $-3$: $12÷(-4)=-3$.
$r=-3$. Each term is multiplied by $-3$: $12÷(-4)=-3$.
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Identify whether $a_n=2n+1$ defines an arithmetic or geometric sequence.
Identify whether $a_n=2n+1$ defines an arithmetic or geometric sequence.
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Arithmetic. Linear form $a_n=2n+1$ has constant differences between consecutive terms.
Arithmetic. Linear form $a_n=2n+1$ has constant differences between consecutive terms.
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Identify whether $a_n=7\cdot 5^{n}$ defines an arithmetic or geometric sequence.
Identify whether $a_n=7\cdot 5^{n}$ defines an arithmetic or geometric sequence.
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Geometric. Exponential form $a_n=7\cdot 5^n$ has constant ratios between consecutive terms.
Geometric. Exponential form $a_n=7\cdot 5^n$ has constant ratios between consecutive terms.
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What is the geometric mean formula that gives the middle positive term between $a$ and $b$ in a geometric sequence?
What is the geometric mean formula that gives the middle positive term between $a$ and $b$ in a geometric sequence?
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$\sqrt{ab}$. Square root of the product of two consecutive positive terms.
$\sqrt{ab}$. Square root of the product of two consecutive positive terms.
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Find $d$ if an arithmetic sequence has $a_n=18-6(n-1)$.
Find $d$ if an arithmetic sequence has $a_n=18-6(n-1)$.
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$d=-6$. The coefficient of $(n-1)$ gives the common difference directly.
$d=-6$. The coefficient of $(n-1)$ gives the common difference directly.
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Find $r$ if a geometric sequence has $a_n=10\cdot\left(\frac{3}{2}\right)^{n-1}$.
Find $r$ if a geometric sequence has $a_n=10\cdot\left(\frac{3}{2}\right)^{n-1}$.
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$r=\frac{3}{2}$. The base of the exponent gives the common ratio directly.
$r=\frac{3}{2}$. The base of the exponent gives the common ratio directly.
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Find $a_7$ for the arithmetic sequence $a_n=2+5(n-1)$.
Find $a_7$ for the arithmetic sequence $a_n=2+5(n-1)$.
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$a_7=32$. Using $a_n=2+5(n-1)$: $a_7=2+5(6)=2+30=32$.
$a_7=32$. Using $a_n=2+5(n-1)$: $a_7=2+5(6)=2+30=32$.
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Find $a_6$ for the geometric sequence $a_n=\left(-2\right)\cdot 3^{n-1}$.
Find $a_6$ for the geometric sequence $a_n=\left(-2\right)\cdot 3^{n-1}$.
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$a_6=-486$. Using $a_n=(-2)\cdot 3^{n-1}$: $a_6=(-2)\cdot 3^5=(-2)\cdot 243=-486$.
$a_6=-486$. Using $a_n=(-2)\cdot 3^{n-1}$: $a_6=(-2)\cdot 3^5=(-2)\cdot 243=-486$.
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Find the arithmetic mean between $8$ and $20$.
Find the arithmetic mean between $8$ and $20$.
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$14$. Using $\frac{a+b}{2}$: $\frac{8+20}{2}=\frac{28}{2}=14$.
$14$. Using $\frac{a+b}{2}$: $\frac{8+20}{2}=\frac{28}{2}=14$.
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Find the geometric mean between $4$ and $36$ (assume positive).
Find the geometric mean between $4$ and $36$ (assume positive).
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$12$. Using $\sqrt{ab}$: $\sqrt{4\cdot 36}=\sqrt{144}=12$.
$12$. Using $\sqrt{ab}$: $\sqrt{4\cdot 36}=\sqrt{144}=12$.
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What is the explicit formula for an arithmetic sequence written from term $a_k$ instead of $a_1$?
What is the explicit formula for an arithmetic sequence written from term $a_k$ instead of $a_1$?
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$a_n=a_k+(n-k)d$. General explicit form starting from any known term $a_k$.
$a_n=a_k+(n-k)d$. General explicit form starting from any known term $a_k$.
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What is the formula for the common ratio $r$ using two terms $a_k$ and $a_m$ of a geometric sequence?
What is the formula for the common ratio $r$ using two terms $a_k$ and $a_m$ of a geometric sequence?
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$r=\left(\frac{a_m}{a_k}\right)^{\frac{1}{m-k}}$. Ratio formula using the $(m-k)$th root of the term quotient.
$r=\left(\frac{a_m}{a_k}\right)^{\frac{1}{m-k}}$. Ratio formula using the $(m-k)$th root of the term quotient.
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What is the explicit formula for a geometric sequence written from term $a_k$ instead of $a_1$?
What is the explicit formula for a geometric sequence written from term $a_k$ instead of $a_1$?
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$a_n=a_k\cdot r^{n-k}$. General explicit form starting from any known term $a_k$.
$a_n=a_k\cdot r^{n-k}$. General explicit form starting from any known term $a_k$.
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Use $a_n=a_k+(n-k)d$: If $a_4=9$ and $d=2$, what is $a_{10}$?
Use $a_n=a_k+(n-k)d$: If $a_4=9$ and $d=2$, what is $a_{10}$?
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$a_{10}=21$. Using the general form: $a_{10}=9+(10-4)(2)=9+12=21$.
$a_{10}=21$. Using the general form: $a_{10}=9+(10-4)(2)=9+12=21$.
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Use $a_n=a_k\cdot r^{n-k}$: If $a_3=16$ and $r=\frac{1}{2}$, what is $a_7$?
Use $a_n=a_k\cdot r^{n-k}$: If $a_3=16$ and $r=\frac{1}{2}$, what is $a_7$?
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$a_7=1$. Using the general form: $a_7=16\cdot\left(\frac{1}{2}\right)^{7-3}=16\cdot\frac{1}{16}=1$.
$a_7=1$. Using the general form: $a_7=16\cdot\left(\frac{1}{2}\right)^{7-3}=16\cdot\frac{1}{16}=1$.
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Find and correct the error: Arithmetic explicit written as $a_n=a_1+nd$.
Find and correct the error: Arithmetic explicit written as $a_n=a_1+nd$.
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Correct: $a_n=a_1+(n-1)d$. The exponent should be $(n-1)$, not $n$, for standard form.
Correct: $a_n=a_1+(n-1)d$. The exponent should be $(n-1)$, not $n$, for standard form.
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