Arithmetic and Geometric Patterns - ISEE Upper Level: Quantitative Reasoning
Card 1 of 24
What is the common ratio $r$ for the geometric sequence $3, -6, 12, -24, \dots$?
What is the common ratio $r$ for the geometric sequence $3, -6, 12, -24, \dots$?
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$r=-2$. Divide consecutive terms to determine the constant ratio of -2 in this geometric sequence.
$r=-2$. Divide consecutive terms to determine the constant ratio of -2 in this geometric sequence.
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Identify whether $7, 12, 17, 22, \dots$ is arithmetic or geometric.
Identify whether $7, 12, 17, 22, \dots$ is arithmetic or geometric.
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Arithmetic. The constant difference of 5 between terms classifies the sequence as arithmetic.
Arithmetic. The constant difference of 5 between terms classifies the sequence as arithmetic.
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Identify whether $5, 15, 45, 135, \dots$ is arithmetic or geometric.
Identify whether $5, 15, 45, 135, \dots$ is arithmetic or geometric.
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Geometric. The constant ratio of 3 between terms identifies the sequence as geometric.
Geometric. The constant ratio of 3 between terms identifies the sequence as geometric.
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What is the next term in the arithmetic sequence $18, 13, 8, 3, \dots$?
What is the next term in the arithmetic sequence $18, 13, 8, 3, \dots$?
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$-2$. Subtract the common difference of -5 from the last term to obtain the next term.
$-2$. Subtract the common difference of -5 from the last term to obtain the next term.
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What is the next term in the geometric sequence $160, 80, 40, 20, \dots$?
What is the next term in the geometric sequence $160, 80, 40, 20, \dots$?
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$10$. Multiply the last term by the common ratio of $\frac{1}{2}$ to find the subsequent term.
$10$. Multiply the last term by the common ratio of $\frac{1}{2}$ to find the subsequent term.
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What is the explicit formula for the $n$th term of an arithmetic sequence with first term $a_1$ and difference $d$?
What is the explicit formula for the $n$th term of an arithmetic sequence with first term $a_1$ and difference $d$?
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$a_n=a_1+(n-1)d$. The formula computes the $n$th term by adding $(n-1)$ times the common difference to the first term.
$a_n=a_1+(n-1)d$. The formula computes the $n$th term by adding $(n-1)$ times the common difference to the first term.
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What is the explicit formula for the $n$th term of a geometric sequence with first term $a_1$ and ratio $r$?
What is the explicit formula for the $n$th term of a geometric sequence with first term $a_1$ and ratio $r$?
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$a_n=a_1\cdot r^{n-1}$. The formula calculates the $n$th term by multiplying the first term by the ratio raised to $(n-1)$.
$a_n=a_1\cdot r^{n-1}$. The formula calculates the $n$th term by multiplying the first term by the ratio raised to $(n-1)$.
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What is $a_{10}$ for the arithmetic sequence with $a_1=4$ and $d=3$?
What is $a_{10}$ for the arithmetic sequence with $a_1=4$ and $d=3$?
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$a_{10}=31$. Substitute $a_1=4$, $d=3$, and $n=10$ into the arithmetic sequence formula to compute the term.
$a_{10}=31$. Substitute $a_1=4$, $d=3$, and $n=10$ into the arithmetic sequence formula to compute the term.
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What is $a_7$ for the geometric sequence with $a_1=2$ and $r=3$?
What is $a_7$ for the geometric sequence with $a_1=2$ and $r=3$?
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$a_7=1458$. Apply the geometric sequence formula with $a_1=2$, $r=3$, and $n=7$ to find the term.
$a_7=1458$. Apply the geometric sequence formula with $a_1=2$, $r=3$, and $n=7$ to find the term.
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Find the missing term in the arithmetic sequence $9, _, 17$.
Find the missing term in the arithmetic sequence $9, _, 17$.
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$13$. The middle term is the arithmetic mean of 9 and 17, as the differences are equal.
$13$. The middle term is the arithmetic mean of 9 and 17, as the differences are equal.
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Find the missing term in the geometric sequence $4, _, 36$.
Find the missing term in the geometric sequence $4, _, 36$.
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$12$. The middle term is the geometric mean of 4 and 36, since ratios are constant.
$12$. The middle term is the geometric mean of 4 and 36, since ratios are constant.
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What is the common ratio $r$ if a geometric sequence has consecutive terms $x$ and $y$?
What is the common ratio $r$ if a geometric sequence has consecutive terms $x$ and $y$?
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$r=\frac{y}{x}$. Divide the second term by the first to obtain the constant ratio in a geometric sequence.
$r=\frac{y}{x}$. Divide the second term by the first to obtain the constant ratio in a geometric sequence.
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What is the common difference $d$ if an arithmetic sequence has consecutive terms $x$ and $y$?
What is the common difference $d$ if an arithmetic sequence has consecutive terms $x$ and $y$?
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$d=y-x$. Subtract the first term from the second to find the constant difference in an arithmetic sequence.
$d=y-x$. Subtract the first term from the second to find the constant difference in an arithmetic sequence.
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Which term equals $0$ in the arithmetic sequence $a_n=12-3(n-1)$?
Which term equals $0$ in the arithmetic sequence $a_n=12-3(n-1)$?
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$n=5$. Set the arithmetic formula equal to 0 and solve for $n$ to identify the term position.
$n=5$. Set the arithmetic formula equal to 0 and solve for $n$ to identify the term position.
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Which term equals $32$ in the geometric sequence $a_n=2\cdot 2^{n-1}$?
Which term equals $32$ in the geometric sequence $a_n=2\cdot 2^{n-1}$?
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$n=5$. Set the geometric formula equal to 32 and solve for $n$ to determine the term number.
$n=5$. Set the geometric formula equal to 32 and solve for $n$ to determine the term number.
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What is the next term in the pattern $1, 4, 9, 16, \dots$?
What is the next term in the pattern $1, 4, 9, 16, \dots$?
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$25$. The pattern consists of squares of consecutive integers: $1^2, 2^2, 3^2, 4^2, 5^2$.
$25$. The pattern consists of squares of consecutive integers: $1^2, 2^2, 3^2, 4^2, 5^2$.
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What is the next term in the pattern $2, 6, 12, 20, \dots$?
What is the next term in the pattern $2, 6, 12, 20, \dots$?
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$30$. Each term follows the pattern $n(n+1)$ for increasing values of $n$ starting from 1.
$30$. Each term follows the pattern $n(n+1)$ for increasing values of $n$ starting from 1.
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What is the next term in the alternating pattern $3, -1, 3, -1, \dots$?
What is the next term in the alternating pattern $3, -1, 3, -1, \dots$?
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$3$. The sequence alternates between 3 and -1, so it repeats 3 after -1.
$3$. The sequence alternates between 3 and -1, so it repeats 3 after -1.
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Identify the pattern type for $1, 1, 2, 3, 5, 8, \dots$ (each term from previous terms).
Identify the pattern type for $1, 1, 2, 3, 5, 8, \dots$ (each term from previous terms).
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Fibonacci-type: $a_n=a_{n-1}+a_{n-2}$. Each term is the sum of the two preceding terms, defining a Fibonacci sequence variant.
Fibonacci-type: $a_n=a_{n-1}+a_{n-2}$. Each term is the sum of the two preceding terms, defining a Fibonacci sequence variant.
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What is the next term in the Fibonacci-type pattern $2, 3, 5, 8, 13, \dots$?
What is the next term in the Fibonacci-type pattern $2, 3, 5, 8, 13, \dots$?
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$21$. Add the last two terms, 8 and 13, to generate the next in this Fibonacci-like sequence.
$21$. Add the last two terms, 8 and 13, to generate the next in this Fibonacci-like sequence.
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Identify whether $0.5, 1, 2, 4, \dots$ is arithmetic or geometric.
Identify whether $0.5, 1, 2, 4, \dots$ is arithmetic or geometric.
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Geometric. A constant ratio of 2 between terms confirms the sequence is geometric.
Geometric. A constant ratio of 2 between terms confirms the sequence is geometric.
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What is the next term in the geometric sequence $-81, 27, -9, 3, \dots$?
What is the next term in the geometric sequence $-81, 27, -9, 3, \dots$?
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$-1$. Multiply the last term by the common ratio of $-\frac{1}{3}$ to find the next term.
$-1$. Multiply the last term by the common ratio of $-\frac{1}{3}$ to find the next term.
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What is the common difference $d$ for the arithmetic sequence $-4, 1, 6, 11, \dots$?
What is the common difference $d$ for the arithmetic sequence $-4, 1, 6, 11, \dots$?
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$d=5$. Subtract consecutive terms to find the constant difference of 5 in this arithmetic sequence.
$d=5$. Subtract consecutive terms to find the constant difference of 5 in this arithmetic sequence.
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What is the next term in the geometric sequence $\frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \frac{1}{16}, \dots$?
What is the next term in the geometric sequence $\frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \frac{1}{16}, \dots$?
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$\frac{1}{32}$. Multiply the last term by the common ratio of $\frac{1}{2}$ in this geometric sequence.
$\frac{1}{32}$. Multiply the last term by the common ratio of $\frac{1}{2}$ in this geometric sequence.
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