ISEE Middle Level Mathematics Achievement Flashcards: Number Pattern Rules

What this deck covers

This deck focuses on Number Pattern Rules, giving you a quick way to review the definitions, rules, and examples that matter most for ISEE Middle Level Mathematics Achievement.

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 rule generates the pattern $2, 5, 10, 17, 26, \dots$?

Answer: Add consecutive odd numbers: $+3,+5,+7,+9,\dots$. The differences between terms are consecutive odd numbers starting from $3$, building the sequence cumulatively.

Flashcard 2: What explicit rule gives the $n$th term of $0, 1, 3, 6, 10, \dots$?

Answer: $a_n = \frac{n(n-1)}{2}$. This formula computes the $(n-1)$th triangular number, summing integers up to $n-1$.

Flashcard 3: What rule generates the pattern $1, 1, 2, 3, 5, 8, \dots$?

Answer: Each term is the sum of the two previous terms. This follows the Fibonacci recurrence, where each term after the first two is the sum of the preceding pair.

Flashcard 4: What explicit rule gives the $n$th term of $2, 5, 10, 17, 26, \dots$?

Answer: $a_n = n^2 + 1$. This formula adds $1$ to the square of $n$ to produce each term in the sequence.

Flashcard 5: What rule generates the pattern $1, 4, 9, 16, 25, \dots$?

Answer: Square the term number: $a_n=n^2$. The pattern consists of perfect squares, where each term is the square of its position $n$.

Flashcard 6: What rule generates the pattern $4, 9, 16, 25, 36, \dots$?

Answer: Square the term number plus $1$: $a_n=(n+1)^2$. Each term is the square of $(n+1)$, producing shifted perfect squares starting from $4$.

Flashcard 7: What explicit rule gives the $n$th term of $64, 32, 16, 8, \dots$?

Answer: $a_n = 64\cdot \left(\frac{1}{2}\right)^{n-1}$. This formula expresses the geometric sequence starting at $64$ with common ratio $\frac{1}{2}$ for the $n$th term.

Flashcard 8: What explicit rule gives the $n$th term of $3, 7, 11, 15, \dots$?

Answer: $a_n = 3 + 4(n-1)$. This formula captures the arithmetic sequence starting at $3$ with a common difference of $4$ to find the $n$th term directly.

Flashcard 9: What rule generates the pattern $1, 2, 4, 7, 11, \dots$?

Answer: Add $1,2,3,4,\dots$. The differences are consecutive positive integers starting from $1$, accumulating to form the sequence.

Flashcard 10: What recursive rule defines $1, 1, 2, 3, 5, 8, \dots$?

Answer: $a_1=1$, $a_2=1$, and $a_n=a_{n-1}+a_{n-2}$. This defines the Fibonacci sequence starting with two $1$s and using the sum of the previous two terms for each subsequent one.

Flashcard 11: What explicit rule gives the $n$th term of $5, 10, 20, 40, \dots$?

Answer: $a_n = 5\cdot 2^{n-1}$. This formula defines the geometric sequence with first term $5$ and common ratio $2$ for the $n$th term.

Flashcard 12: What rule generates the pattern $2, 6, 12, 20, 30, \dots$?

Answer: Products of consecutive integers: $a_n=n(n+1)$. Each term is the product of $n$ and $(n+1)$, generating twice the triangular numbers.

Flashcard 13: What explicit rule gives the $n$th term of $3, -3, 3, -3, \dots$?

Answer: $a_n = 3(-1)^{n-1}$. The formula uses powers of $-1$ to alternate signs, starting positive for odd $n$ and negative for even $n$.

Flashcard 14: What explicit rule gives the $n$th term of $7, 12, 17, 22, \dots$?

Answer: $a_n = 7 + 5(n-1)$. This arithmetic formula uses the first term $7$ and common difference $5$ to find the $n$th term.

Flashcard 15: What explicit rule gives the $n$th term of $20, 16, 12, 8, \dots$?

Answer: $a_n = 20 - 4(n-1)$. This formula represents the arithmetic sequence starting at $20$ with a common difference of $-4$ for the $n$th term.

Flashcard 16: What recursive rule generates $2, 6, 18, 54, \dots$?

Answer: $a_1=2$, and $a_n=3a_{n-1}$. This recursive definition starts with $2$ and multiplies each subsequent term by $3$, forming a geometric sequence.

Flashcard 17: What rule generates the arithmetic pattern $5, 9, 13, 17, \dots$?

Answer: Add $4$ each term. The pattern is an arithmetic sequence with a common difference of $4$, increasing each term by that constant.

Flashcard 18: What rule generates the alternating pattern $2, 5, 2, 5, 2, \dots$?

Answer: Repeat the cycle $2,5$. The pattern is periodic with period $2$, alternating between $2$ and $5$ indefinitely.

Flashcard 19: What rule generates the arithmetic pattern $18, 15, 12, 9, \dots$?

Answer: Subtract $3$ each term. The pattern is an arithmetic sequence with a common difference of $-3$, decreasing each term by $3$.

Flashcard 20: What explicit rule gives the $n$th term of $1, 2, 4, 7, 11, \dots$?

Answer: $a_n = 1 + \frac{n(n-1)}{2}$. This formula adds $1$ to the $(n-1)$th triangular number to generate each term.

Flashcard 21: What rule generates the pattern $1, 3, 6, 10, 15, \dots$?

Answer: Add $2,3,4,5,\dots$. The differences start from $2$ and increase by $1$ each time, forming triangular numbers beginning at $1$.

Flashcard 22: What rule generates the pattern $0, 1, 3, 6, 10, \dots$?

Answer: Add $1,2,3,4,\dots$ (triangular numbers). The pattern forms triangular numbers by cumulatively adding consecutive positive integers starting from $1$.

Flashcard 23: What rule generates the geometric pattern $81, 27, 9, 3, \dots$?

Answer: Multiply by $\frac{1}{3}$ each term. The pattern is a geometric sequence with a common ratio of $\frac{1}{3}$, reducing each term accordingly.

Flashcard 24: What rule generates the pattern $1, 8, 27, 64, \dots$?

Answer: Cube the term number: $a_n=n^3$. The pattern consists of perfect cubes, where each term is the cube of its position $n$.

Flashcard 25: What rule generates the alternating pattern $3, -3, 3, -3, \dots$?

Answer: Multiply by $-1$ each term. Multiplying by $-1$ alternates the sign of each term while keeping the absolute value constant at $3$.