Arithmetic and Geometric Sequences as Functions - Algebra
Card 1 of 30
Identify whether $-1, 2, 5, 8, \dots$ is arithmetic or geometric.
Identify whether $-1, 2, 5, 8, \dots$ is arithmetic or geometric.
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Arithmetic. Differences are constant: $2-(-1) = 3$.
Arithmetic. Differences are constant: $2-(-1) = 3$.
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Identify the error: A student writes geometric explicit form as $a_n = a_1 r^n$. What is correct?
Identify the error: A student writes geometric explicit form as $a_n = a_1 r^n$. What is correct?
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Correct: $a_n = a_1 r^{n-1}$. The exponent should be $(n-1)$, not $n$.
Correct: $a_n = a_1 r^{n-1}$. The exponent should be $(n-1)$, not $n$.
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Identify the error: A student writes arithmetic explicit form as $a_n = a_1 + nd$. What is correct?
Identify the error: A student writes arithmetic explicit form as $a_n = a_1 + nd$. What is correct?
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Correct: $a_n = a_1 + (n-1)d$. The exponent should be $(n-1)$, not $n$.
Correct: $a_n = a_1 + (n-1)d$. The exponent should be $(n-1)$, not $n$.
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Find the missing term to make $4, x, 16$ a geometric sequence with positive ratio.
Find the missing term to make $4, x, 16$ a geometric sequence with positive ratio.
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$x = 8$. Use geometric mean: $x = \sqrt{4 \cdot 16} = 8$.
$x = 8$. Use geometric mean: $x = \sqrt{4 \cdot 16} = 8$.
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Find the missing term to make $4, x, 16$ an arithmetic sequence.
Find the missing term to make $4, x, 16$ an arithmetic sequence.
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$x = 10$. Use arithmetic mean: $x = \frac{4 + 16}{2} = 10$.
$x = 10$. Use arithmetic mean: $x = \frac{4 + 16}{2} = 10$.
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Model: A car value is $\$20{,}000$ and keeps $0.85$ of its value yearly. Write $V_n$ explicitly.
Model: A car value is $\$20{,}000$ and keeps $0.85$ of its value yearly. Write $V_n$ explicitly.
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$V_n = 20000\left(0.85\right)^{n-1}$. Geometric sequence models exponential decay with factor $0.85$.
$V_n = 20000\left(0.85\right)^{n-1}$. Geometric sequence models exponential decay with factor $0.85$.
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What is the geometric mean of two positive numbers $x$ and $y$ (the middle term in a geometric sequence)?
What is the geometric mean of two positive numbers $x$ and $y$ (the middle term in a geometric sequence)?
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$\sqrt{xy}$. The geometric mean is the square root of the product of two positive numbers.
$\sqrt{xy}$. The geometric mean is the square root of the product of two positive numbers.
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Model: A population starts at $500$ and grows by a factor of $1.08$ yearly. What is $P_n$?
Model: A population starts at $500$ and grows by a factor of $1.08$ yearly. What is $P_n$?
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$P_n = 500\left(1.08\right)^{n-1}$. Geometric sequence models constant growth factor.
$P_n = 500\left(1.08\right)^{n-1}$. Geometric sequence models constant growth factor.
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Model: A salary starts at $\$40{,}000$ and increases by $$1{,}500$ yearly. What is $S_n$?
Model: A salary starts at $\$40{,}000$ and increases by $$1{,}500$ yearly. What is $S_n$?
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$S_n = 40000 + 1500(n-1)$. Arithmetic sequence models constant yearly increases.
$S_n = 40000 + 1500(n-1)$. Arithmetic sequence models constant yearly increases.
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Find $a_6$ for the geometric sequence $a_1 = 1$ and $r = \frac{1}{3}$.
Find $a_6$ for the geometric sequence $a_1 = 1$ and $r = \frac{1}{3}$.
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$a_6 = \frac{1}{243}$. Use $a_n = a_1 r^{n-1} = 1 \cdot \left(\frac{1}{3}\right)^5 = \frac{1}{243}$.
$a_6 = \frac{1}{243}$. Use $a_n = a_1 r^{n-1} = 1 \cdot \left(\frac{1}{3}\right)^5 = \frac{1}{243}$.
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Find $a_8$ for the arithmetic sequence $a_1 = -5$ and $d = 2$.
Find $a_8$ for the arithmetic sequence $a_1 = -5$ and $d = 2$.
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$a_8 = 9$. Use $a_n = a_1 + (n-1)d = -5 + 7(2) = 9$.
$a_8 = 9$. Use $a_n = a_1 + (n-1)d = -5 + 7(2) = 9$.
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Write an explicit formula for the geometric sequence with $a_1 = -3$ and $r = -2$.
Write an explicit formula for the geometric sequence with $a_1 = -3$ and $r = -2$.
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$a_n = -3\left(-2\right)^{n-1}$. Use explicit form $a_n = a_1 r^{n-1}$ with given values.
$a_n = -3\left(-2\right)^{n-1}$. Use explicit form $a_n = a_1 r^{n-1}$ with given values.
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Write a recursive formula for the arithmetic sequence defined by $a_n = -2n + 9$.
Write a recursive formula for the arithmetic sequence defined by $a_n = -2n + 9$.
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$a_1 = 7$; $a_n = a_{n-1} - 2$. Convert $a_n = -2n + 9$ to recursive by finding $a_1 = 7$ and $d = -2$.
$a_1 = 7$; $a_n = a_{n-1} - 2$. Convert $a_n = -2n + 9$ to recursive by finding $a_1 = 7$ and $d = -2$.
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Find $r$ for a geometric sequence with $a_2 = 6$ and $a_5 = 162$.
Find $r$ for a geometric sequence with $a_2 = 6$ and $a_5 = 162$.
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$r = 3$. Use $\frac{a_5}{a_2} = r^3$ to solve: $\frac{162}{6} = 27 = r^3$.
$r = 3$. Use $\frac{a_5}{a_2} = r^3$ to solve: $\frac{162}{6} = 27 = r^3$.
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Find $d$ for an arithmetic sequence with $a_2 = 9$ and $a_5 = 21$.
Find $d$ for an arithmetic sequence with $a_2 = 9$ and $a_5 = 21$.
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$d = 4$. Use $a_5 - a_2 = 3d$ to solve: $21 - 9 = 12 = 3d$.
$d = 4$. Use $a_5 - a_2 = 3d$ to solve: $21 - 9 = 12 = 3d$.
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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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$a_n = a_1 + (n-1)d$. This formula adds $(n-1)$ multiples of the common difference $d$ to the first term.
$a_n = a_1 + (n-1)d$. This formula adds $(n-1)$ multiples of the common difference $d$ to the first term.
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What is the recursive formula for an arithmetic sequence with first term $a_1$ and common difference $d$?
What is the recursive formula for an arithmetic sequence with first term $a_1$ and common difference $d$?
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$a_1$ given; $a_n = a_{n-1} + d$. Each term equals the previous term plus the common difference $d$.
$a_1$ given; $a_n = a_{n-1} + d$. Each term equals the previous term plus the common difference $d$.
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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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$a_n = a_1 r^{n-1}$. This formula multiplies the first term by $r$ raised to the $(n-1)$th power.
$a_n = a_1 r^{n-1}$. This formula multiplies the first term by $r$ raised to the $(n-1)$th power.
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What is the recursive formula for a geometric sequence with first term $a_1$ and common ratio $r$?
What is the recursive formula for a geometric sequence with first term $a_1$ and common ratio $r$?
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$a_1$ given; $a_n = r a_{n-1}$. Each term equals the previous term multiplied by the common ratio $r$.
$a_1$ given; $a_n = r a_{n-1}$. Each term equals the previous term multiplied by the common ratio $r$.
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What is the common difference $d$ in an arithmetic sequence in terms of consecutive terms $a_n$ and $a_{n-1}$?
What is the common difference $d$ in an arithmetic sequence in terms of consecutive terms $a_n$ and $a_{n-1}$?
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$d = a_n - a_{n-1}$. The common difference is found by subtracting consecutive terms.
$d = a_n - a_{n-1}$. The common difference is found by subtracting consecutive terms.
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What is the common ratio $r$ in a geometric sequence in terms of consecutive terms $a_n$ and $a_{n-1}$?
What is the common ratio $r$ in a geometric sequence in terms of consecutive terms $a_n$ and $a_{n-1}$?
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$r = \frac{a_n}{a_{n-1}}$. The common ratio is found by dividing consecutive terms.
$r = \frac{a_n}{a_{n-1}}$. The common ratio is found by dividing consecutive terms.
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What condition on consecutive differences identifies an arithmetic sequence?
What condition on consecutive differences identifies an arithmetic sequence?
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All $a_n - a_{n-1}$ are equal. Constant differences between consecutive terms characterize arithmetic sequences.
All $a_n - a_{n-1}$ are equal. Constant differences between consecutive terms characterize arithmetic sequences.
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What condition on consecutive ratios identifies a geometric sequence?
What condition on consecutive ratios identifies a geometric sequence?
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All $\frac{a_n}{a_{n-1}}$ are equal. Constant ratios between consecutive terms characterize geometric sequences.
All $\frac{a_n}{a_{n-1}}$ are equal. Constant ratios between consecutive terms characterize geometric sequences.
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What is the arithmetic mean of two numbers $x$ and $y$ (the middle term in an arithmetic sequence)?
What is the arithmetic mean of two numbers $x$ and $y$ (the middle term in an arithmetic sequence)?
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$\frac{x+y}{2}$. The arithmetic mean is the average of two numbers.
$\frac{x+y}{2}$. The arithmetic mean is the average of two numbers.
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Given $a_4 = 54$ and $r = 3$ for a geometric sequence, find $a_1$.
Given $a_4 = 54$ and $r = 3$ for a geometric sequence, find $a_1$.
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$a_1 = 2$. Use $a_4 = a_1 r^3$ to solve: $54 = a_1 \cdot 27$.
$a_1 = 2$. Use $a_4 = a_1 r^3$ to solve: $54 = a_1 \cdot 27$.
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Given $a_3 = 11$ and $d = 4$ for an arithmetic sequence, find $a_1$.
Given $a_3 = 11$ and $d = 4$ for an arithmetic sequence, find $a_1$.
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$a_1 = 3$. Use $a_3 = a_1 + 2d$ to solve: $11 = a_1 + 8$.
$a_1 = 3$. Use $a_3 = a_1 + 2d$ to solve: $11 = a_1 + 8$.
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Find the next term in the geometric sequence $160, 80, 40, 20, \dots$.
Find the next term in the geometric sequence $160, 80, 40, 20, \dots$.
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$10$. Multiply the last term $20$ by the common ratio $r = \frac{1}{2}$.
$10$. Multiply the last term $20$ by the common ratio $r = \frac{1}{2}$.
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Find the next term in the arithmetic sequence $-4, -1, 2, 5, \dots$.
Find the next term in the arithmetic sequence $-4, -1, 2, 5, \dots$.
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$8$. Add the common difference $d = 3$ to the last term $5$.
$8$. Add the common difference $d = 3$ to the last term $5$.
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Identify whether $3, -6, 12, -24, \dots$ is arithmetic or geometric.
Identify whether $3, -6, 12, -24, \dots$ is arithmetic or geometric.
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Geometric. Ratios are constant: $\frac{-6}{3} = -2$.
Geometric. Ratios are constant: $\frac{-6}{3} = -2$.
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Identify whether $-1, 2, 5, 8, \dots$ is arithmetic or geometric.
Identify whether $-1, 2, 5, 8, \dots$ is arithmetic or geometric.
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Arithmetic. Differences are constant: $2-(-1) = 3$.
Arithmetic. Differences are constant: $2-(-1) = 3$.
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