ISEE Lower 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 Lower 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 is the rule for $1, 3, 6, 10, 15, \dots$ (state $a_n$ in terms of $n$)?

Answer: $a_n = \frac{n(n+1)}{2}$. Triangular numbers arise from the sum of first $n$ positives, given by this quadratic formula.

Flashcard 2: Identify the rule for $1, 1, 2, 3, 5, 8, \dots$ (state the recursive rule).

Answer: $a_1=1,\ a_2=1,\ a_n=a_{n-1}+a_{n-2}$. The Fibonacci sequence is defined recursively by summing the two preceding terms after initial values.

Flashcard 3: Identify the rule for $64, 32, 16, 8, 4, \dots$ (state $a_n$ in terms of $n$).

Answer: $a_n = 64\left(\frac{1}{2}\right)^{n-1}$. Geometric sequence starting at 64 with ratio $\frac{1}{2}$, following the explicit geometric form.

Flashcard 4: Which option is the common difference for the arithmetic sequence $-4, -1, 2, 5, \dots$?

Answer: $3$. The constant difference between consecutive terms in this arithmetic sequence is 3.

Flashcard 5: What is the next term if the rule is $a_n = 7 + 2(n-1)$ and $a_4$ is the last shown term?

Answer: $a_5 = 15$. Applying the arithmetic rule for $n=5$ gives the subsequent term after $a_4$.

Flashcard 6: Identify the rule for $5, 10, 20, 40, 80, \dots$ (state $a_n$ in terms of $n$).

Answer: $a_n = 5\cdot 2^{n-1}$. Geometric sequence with initial 5 and ratio 2, using the standard explicit formula.

Flashcard 7: What is the rule for $1, 3, 5, 7, 9, \dots$ (state $a_n$ in terms of $n$)?

Answer: $a_n = 2n - 1$. This represents odd numbers, derived as twice the index minus 1.

Flashcard 8: What is the rule for the alternating sequence $-2, 2, -2, 2, \dots$ (state $a_n$)?

Answer: $a_n = 2(-1)^n$. The pattern alternates signs with magnitude 2, achieved by scaling $(-1)^n$.

Flashcard 9: Identify the rule for $81, 27, 9, 3, 1, \dots$ (state $a_n$ in terms of $n$).

Answer: $a_n = 3^{5-n}$. This geometric sequence starts at 81 with common ratio $\frac{1}{3}$, expressible as decreasing powers of 3.

Flashcard 10: What is the rule for the sequence $12, 9, 6, 3, 0, \dots$ (state $a_n$ in terms of $n$)?

Answer: $a_n = 15 - 3n$. This arithmetic sequence has first term 12 and common difference -3, derived from $a_n = a_1 + (n-1)d$.

Flashcard 11: Identify the rule for $2, 5, 10, 17, 26, \dots$ (state $a_n$ in terms of $n$).

Answer: $a_n = n^2 + 1$. Each term adds 1 to the square of its position, creating a quadratic pattern.

Flashcard 12: What is the rule for the sequence $3, 7, 11, 15, \dots$ (state $a_n$ in terms of $n$)?

Answer: $a_n = 4n - 1$. The sequence is arithmetic with first term 3 and common difference 4, yielding the general term through the formula $a_n = a_1 + (n-1)d$.

Flashcard 13: Identify the rule for $48, 16, \frac{16}{3}, \frac{16}{9}, \dots$ (give the recursive rule).

Answer: $a_1=48,\ a_n=\frac{1}{3}a_{n-1}$. Geometric sequence multiplies each term by $\frac{1}{3}$ starting from 48.

Flashcard 14: What is the explicit formula for an arithmetic sequence with $a_1=9$ and common difference $d=-4$?

Answer: $a_n = 9 - 4(n-1)$. The explicit form for an arithmetic sequence uses first term and common difference in $a_n = a_1 + (n-1)d$.

Flashcard 15: What is the rule for $2, 6, 12, 20, 30, \dots$ (state $a_n$ in terms of $n$)?

Answer: $a_n = n(n+1)$. Each term is the product of consecutive integers, forming a quadratic sequence.

Flashcard 16: What is the explicit formula for a geometric sequence with $a_1=6$ and common ratio $r=\frac{1}{2}$?

Answer: $a_n = 6\left(\frac{1}{2}\right)^{n-1}$. The explicit geometric formula incorporates first term and common ratio as $a_n = a_1 r^{n-1}$.

Flashcard 17: Identify the rule for $2, 4, 8, 16, 32, \dots$ (state $a_n$ in terms of $n$).

Answer: $a_n = 2^n$. The sequence is geometric with first term 2 and common ratio 2, simplifying to powers of 2.

Flashcard 18: Identify the rule for $1, 2, 4, 7, 11, \dots$ (give the recursive rule).

Answer: $a_1=1,\ a_n=a_{n-1}+(n-1)$. Each term adds the previous index value, creating increasing differences.

Flashcard 19: Which option is the common ratio for the geometric sequence $3, -6, 12, -24, \dots$?

Answer: $-2$. The constant ratio in this geometric sequence, alternating signs, is -2.

Flashcard 20: Identify the rule for $0, 1, 4, 9, 16, \dots$ (state $a_n$ in terms of $n$).

Answer: $a_n = (n-1)^2$. The sequence shifts perfect squares by starting at 0 for $n=1$.

Flashcard 21: What is the rule for $1, 4, 9, 16, 25, \dots$ (state $a_n$ in terms of $n$)?

Answer: $a_n = n^2$. Each term is the square of its position index, forming the sequence of perfect squares.

Flashcard 22: What is the rule for $2, 4, 6, 8, 10, \dots$ (state $a_n$ in terms of $n$)?

Answer: $a_n = 2n$. Even numbers are generated by multiplying the index by 2.

Flashcard 23: What is the rule for $1, 8, 27, 64, 125, \dots$ (state $a_n$ in terms of $n$)?

Answer: $a_n = n^3$. Terms are cubes of consecutive integers starting from 1.

Flashcard 24: What is the rule for the alternating sequence $1, -1, 1, -1, \dots$ (state $a_n$)?

Answer: $a_n = (-1)^{n-1}$. Alternating signs are produced by raising -1 to an exponent that toggles parity.

Flashcard 25: Identify the rule for $2, 5, 8, 11, 14, \dots$ (give the recursive rule).

Answer: $a_1=2,\ a_n=a_{n-1}+3$. This arithmetic sequence adds a constant difference of 3 to each previous term starting from 2.