ISEE Upper Level Quantitative Reasoning Flashcards: Arithmetic And Geometric Patterns

What this deck covers

This deck focuses on Arithmetic And Geometric Patterns, giving you a quick way to review the definitions, rules, and examples that matter most for ISEE Upper Level Quantitative Reasoning.

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 is the common ratio $r$ for the geometric sequence $3, -6, 12, -24, \dots$?

Answer: $r=-2$. Divide consecutive terms to determine the constant ratio of -2 in this geometric sequence.

Flashcard 2: What is the next term in the arithmetic sequence $18, 13, 8, 3, \dots$?

Answer: $-2$. Subtract the common difference of -5 from the last term to obtain the next term.

Flashcard 3: What is the common difference $d$ for the arithmetic sequence $-4, 1, 6, 11, \dots$?

Answer: $d=5$. Subtract consecutive terms to find the constant difference of 5 in this arithmetic sequence.

Flashcard 4: Which term equals $32$ in the geometric sequence $a_n=2\cdot 2^{n-1}$?

Answer: $n=5$. Set the geometric formula equal to 32 and solve for $n$ to determine the term number.

Flashcard 5: What is the explicit formula for the $n$th term of a geometric sequence with first term $a_1$ and ratio $r$?

Answer: $a_n=a_1\cdot r^{n-1}$. The formula calculates the $n$th term by multiplying the first term by the ratio raised to $(n-1)$.

Flashcard 6: What is the next term in the geometric sequence $160, 80, 40, 20, \dots$?

Answer: $10$. Multiply the last term by the common ratio of $\frac{1}{2}$ to find the subsequent term.

Flashcard 7: What is the next term in the geometric sequence $\frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \frac{1}{16}, \dots$?

Answer: $\frac{1}{32}$. Multiply the last term by the common ratio of $\frac{1}{2}$ in this geometric sequence.

Flashcard 8: What is the next term in the Fibonacci-type pattern $2, 3, 5, 8, 13, \dots$?

Answer: $21$. Add the last two terms, 8 and 13, to generate the next in this Fibonacci-like sequence.

Flashcard 9: Identify whether $0.5, 1, 2, 4, \dots$ is arithmetic or geometric.

Answer: Geometric. A constant ratio of 2 between terms confirms the sequence is geometric.

Flashcard 10: Identify the pattern type for $1, 1, 2, 3, 5, 8, \dots$ (each term from previous terms).

Answer: 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.

Flashcard 11: What is the next term in the pattern $2, 6, 12, 20, \dots$?

Answer: $30$. Each term follows the pattern $n(n+1)$ for increasing values of $n$ starting from 1.

Flashcard 12: What is the next term in the geometric sequence $-81, 27, -9, 3, \dots$?

Answer: $-1$. Multiply the last term by the common ratio of $-\frac{1}{3}$ to find the next term.

Flashcard 13: What is the common difference $d$ if an arithmetic sequence has consecutive terms $x$ and $y$?

Answer: $d=y-x$. Subtract the first term from the second to find the constant difference in an arithmetic sequence.

Flashcard 14: What is the next term in the pattern $1, 4, 9, 16, \dots$?

Answer: $25$. The pattern consists of squares of consecutive integers: $1^2, 2^2, 3^2, 4^2, 5^2$.

Flashcard 15: Find the missing term in the geometric sequence $4, _, 36$.

Answer: $12$. The middle term is the geometric mean of 4 and 36, since ratios are constant.

Flashcard 16: Identify whether $5, 15, 45, 135, \dots$ is arithmetic or geometric.

Answer: Geometric. The constant ratio of 3 between terms identifies the sequence as geometric.

Flashcard 17: What is the next term in the alternating pattern $3, -1, 3, -1, \dots$?

Answer: $3$. The sequence alternates between 3 and -1, so it repeats 3 after -1.

Flashcard 18: Which term equals $0$ in the arithmetic sequence $a_n=12-3(n-1)$?

Answer: $n=5$. Set the arithmetic formula equal to 0 and solve for $n$ to identify the term position.

Flashcard 19: Find the missing term in the arithmetic sequence $9, _, 17$.

Answer: $13$. The middle term is the arithmetic mean of 9 and 17, as the differences are equal.

Flashcard 20: What is $a_7$ for the geometric sequence with $a_1=2$ and $r=3$?

Answer: $a_7=1458$. Apply the geometric sequence formula with $a_1=2$, $r=3$, and $n=7$ to find the term.

Flashcard 21: Identify whether $7, 12, 17, 22, \dots$ is arithmetic or geometric.

Answer: Arithmetic. The constant difference of 5 between terms classifies the sequence as arithmetic.

Flashcard 22: What is $a_{10}$ for the arithmetic sequence with $a_1=4$ and $d=3$?

Answer: $a_{10}=31$. Substitute $a_1=4$, $d=3$, and $n=10$ into the arithmetic sequence formula to compute the term.

Flashcard 23: What is the common ratio $r$ if a geometric sequence has consecutive terms $x$ and $y$?

Answer: $r=\frac{y}{x}$. Divide the second term by the first to obtain the constant ratio in a geometric sequence.

Flashcard 24: What is the explicit formula for the $n$th term of an arithmetic sequence with first term $a_1$ and difference $d$?

Answer: $a_n=a_1+(n-1)d$. The formula computes the $n$th term by adding $(n-1)$ times the common difference to the first term.