ISEE Middle Level Quantitative Reasoning Flashcards: Pattern Rules

What this deck covers

This deck focuses on Pattern Rules, giving you a quick way to review the definitions, rules, and examples that matter most for ISEE Middle 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 rule for the sequence $\frac{1}{2}, 1, 2, 4, \dots$ in terms of $n$?

Answer: $a_n = \frac{1}{2}\cdot 2^{n-1}$. The geometric sequence begins with $\frac{1}{2}$ and has a common ratio of 2, forming the explicit rule.

Flashcard 2: Identify the next term in the sequence $81, 27, 9, 3, 1, \dots$.

Answer: $\frac{1}{3}$. The geometric sequence has a common ratio of $\frac{1}{3}$, so multiply the last term by $\frac{1}{3}$.

Flashcard 3: What is the rule for the sequence $2, 6, 18, 54, \dots$ in terms of $n$?

Answer: $a_n = 2\cdot 3^{n-1}$. The geometric sequence starts with 2 and has a common ratio of 3, forming the rule.

Flashcard 4: Identify the next term in the sequence $12, 11, 9, 6, 2, \dots$.

Answer: $-3$. The differences are -1, -2, -3, -4, decreasing by 1 each time, so subtract 5 from 2.

Flashcard 5: Identify the next term in the sequence $2, 5, 10, 17, 26, \dots$.

Answer: $37$. The differences between terms are 3, 5, 7, 9, increasing by 2 each time, so the next difference is 11 added to 26.

Flashcard 6: Identify the next term in the sequence $5, 15, 45, 135, \dots$.

Answer: $405$. The geometric sequence has a common ratio of 3, so multiply 135 by 3.

Flashcard 7: What is the rule for the sequence $0, 1, 4, 9, 16, \dots$ using $n$?

Answer: $a_n = (n-1)^2$. Each term is the square of one less than its position number.

Flashcard 8: What is the rule for the sequence $10, 7, 4, 1, -2, \dots$ in terms of $n$?

Answer: $a_n = 10 - 3(n-1)$. The sequence is arithmetic with first term 10 and common difference -3, yielding the explicit formula.

Flashcard 9: What is the common ratio for the sequence $160, 80, 40, 20, \dots$?

Answer: $r = \frac{1}{2}$. Each term is half of the previous, defining the constant ratio.

Flashcard 10: What is the rule for the geometric sequence $3, 6, 12, 24, \dots$ in terms of $n$?

Answer: $a_n = 3\cdot 2^{n-1}$. The sequence is geometric with first term 3 and common ratio 2, forming the explicit rule.

Flashcard 11: What is the pattern rule for $1, 4, 9, 16, 25, \dots$ using $n$?

Answer: $a_n = n^2$. Each term is the square of its position number in the sequence.

Flashcard 12: What is the common difference for the sequence $-3, 2, 7, 12, \dots$?

Answer: $d = 5$. Each term increases by 5 from the previous, establishing the constant difference.

Flashcard 13: Identify the next term in the sequence $4, 7, 13, 25, 49, \dots$.

Answer: $97$. Each term is obtained by doubling the previous term and subtracting 1.

Flashcard 14: What is the explicit rule for the arithmetic sequence $-8, -3, 2, 7, \dots$?

Answer: $a_n = -8 + 5(n-1)$. The arithmetic sequence has first term -8 and common difference 5, leading to the explicit rule.

Flashcard 15: What is the rule for the sequence $7, 14, 21, 28, \dots$ in terms of $n$?

Answer: $a_n = 7n$. The sequence consists of multiples of 7, directly proportional to $n$.

Flashcard 16: Identify the next term in the sequence $1, 3, 6, 10, 15, \dots$.

Answer: $21$. These are triangular numbers where each term is the sum of the first $n$ natural numbers, and the differences increase by 1, so add 6 to 15.

Flashcard 17: Identify the next term in the sequence $2, 4, 7, 11, 16, \dots$.

Answer: $22$. The differences are 2, 3, 4, 5, increasing by 1 each time, so add 6 to 16.

Flashcard 18: Identify the next term in the sequence $2, 3, 5, 9, 17, \dots$.

Answer: $33$. The differences are 1, 2, 4, 8, doubling each time, so add 16 to 17.

Flashcard 19: What is the rule for the sequence $1, 8, 27, 64, 125, \dots$ using $n$?

Answer: $a_n = n^3$. Each term is the cube of its position number in the sequence.

Flashcard 20: What is the rule for the arithmetic sequence $5, 9, 13, 17, \dots$ in terms of $n$?

Answer: $a_n = 5 + 4(n-1)$. The sequence is arithmetic with the first term 5 and common difference 4, so the explicit formula uses these values.

Flashcard 21: Identify the next term in the sequence $1, 1, 2, 3, 5, 8, \dots$.

Answer: $13$. This is the Fibonacci sequence where each term is the sum of the two preceding ones.

Flashcard 22: What is the explicit rule for the sequence $2, 1, \frac{1}{2}, \frac{1}{4}, \dots$?

Answer: $a_n = 2\cdot \left(\frac{1}{2}\right)^{n-1}$. The geometric sequence starts with 2 and has a common ratio of $\frac{1}{2}$, yielding the explicit formula.

Flashcard 23: Identify the next term in the sequence $3, 6, 10, 15, 21, \dots$.

Answer: $28$. The differences are 3, 4, 5, 6, increasing by 1, so add 7 to 21.

Flashcard 24: What is the rule for the sequence $9, 6, 4, \frac{8}{3}, \dots$ in terms of $n$?

Answer: $a_n = 9\cdot \left(\frac{2}{3}\right)^{n-1}$. The geometric sequence has first term 9 and common ratio $\frac{2}{3}$, forming the rule.