Sequences and patterns
49 flashcards covering Sequences and patterns for the ACT Math section.
Sequences and patterns are fundamental concepts in math that involve ordered lists of numbers or shapes following a specific rule. For example, a sequence might be 3, 6, 9, 12, where each number increases by 3, or it could be a geometric pattern like 2, 4, 8, 16, doubling each time. These ideas help build skills in recognizing rules, predicting future terms, and solving problems with repetition, which are essential for understanding more complex math topics.
On the ACT Math section, sequences and patterns often appear in questions asking you to find the nth term, identify the next number in a series, or solve word problems involving growth or repetition. Common traps include overlooking alternating patterns or confusing arithmetic sequences (with a constant difference) with geometric ones (with a constant ratio), so double-check the rule before calculating. Focus on mastering formulas like the sum of an arithmetic series and practicing quick pattern recognition to save time during the test.
Always practice with mixed problem sets to spot sequences faster.
Terms (49)
- 01
Sequence
A sequence is an ordered list of numbers that follows a particular pattern or rule.
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Arithmetic Sequence
An arithmetic sequence is a sequence of numbers where the difference between consecutive terms is constant.
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Geometric Sequence
A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous one by a constant called the common ratio.
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Common Difference
The common difference in an arithmetic sequence is the fixed amount added to each term to get the next term.
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Common Ratio
The common ratio in a geometric sequence is the fixed number multiplied by each term to get the next term.
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nth Term of an Arithmetic Sequence
The nth term of an arithmetic sequence is given by the formula an = a1 + (n-1)d, where a1 is the first term, d is the common difference, and n is the term number.
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nth Term of a Geometric Sequence
The nth term of a geometric sequence is given by the formula an = a1 r^(n-1), where a1 is the first term, r is the common ratio, and n is the term number.
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Sum of the First n Terms of an Arithmetic Sequence
The sum of the first n terms of an arithmetic sequence is given by Sn = n/2 (a1 + an), or Sn = n/2 [2a1 + (n-1)d].
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Sum of the First n Terms of a Geometric Sequence
The sum of the first n terms of a geometric sequence is given by Sn = a1 (1 - r^n) / (1 - r), where r is the common ratio not equal to 1.
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Sum of an Infinite Geometric Series
The sum of an infinite geometric series is S = a1 / (1 - r), provided that the absolute value of the common ratio r is less than 1.
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Recursive Formula for a Sequence
A recursive formula for a sequence defines each term based on the previous term, such as an = a{n-1} + d for an arithmetic sequence.
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Explicit Formula for a Sequence
An explicit formula for a sequence allows you to find the nth term directly without knowing the previous terms, like an = a1 + (n-1)d for arithmetic sequences.
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First Term of a Sequence
The first term of a sequence is the initial number in the ordered list, often denoted as a1 or a0.
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Number of Terms in a Sequence
The number of terms in a sequence can be found by determining how many values satisfy the sequence's pattern up to a given last term.
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Constant Sequence
A constant sequence is one where every term is the same number, such as 5, 5, 5, 5.
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Increasing Sequence
An increasing sequence is one where each term is larger than the previous one.
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Decreasing Sequence
A decreasing sequence is one where each term is smaller than the previous one.
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Alternating Sequence
An alternating sequence is one where the terms switch between increasing and decreasing or positive and negative in a pattern.
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Fibonacci Sequence
The Fibonacci sequence is a series where each term is the sum of the two preceding ones, starting with 0 and 1, such as 0, 1, 1, 2, 3, 5.
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Triangular Numbers
Triangular numbers are a sequence where each number is the sum of the first n natural numbers, like 1, 3, 6, 10, representing the number of dots in an equilateral triangle.
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Pattern Recognition in Sequences
Pattern recognition in sequences involves identifying the rule that governs how terms are generated, such as addition or multiplication.
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Sigma Notation
Sigma notation is a way to write the sum of a sequence using the symbol Σ, such as Σ from i=1 to n of ai for the sum of the first n terms.
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Arithmetic Mean of a Sequence
The arithmetic mean of a sequence is the average of its terms, calculated by summing all terms and dividing by the number of terms.
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Geometric Mean of a Sequence
The geometric mean of a sequence is the nth root of the product of n terms, often used in geometric sequences.
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Strategy for Finding the nth Term
To find the nth term, first identify if the sequence is arithmetic or geometric, then use the appropriate formula with the first term and common difference or ratio.
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Common Trap: Off-by-One Error in Sequences
An off-by-one error occurs when miscounting the term number, such as forgetting that the first term is n=1, not n=0.
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Common Trap: Confusing Arithmetic and Geometric Sequences
Confusing arithmetic and geometric sequences happens when you add instead of multiply, or vice versa, for the next term.
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Example: Arithmetic Sequence with First Term 2 and Common Difference 3
In an arithmetic sequence starting with 2 and a common difference of 3, the terms are 2, 5, 8, 11, and so on.
The fourth term is 2 + 3(4-1) = 11.
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Example: Geometric Sequence with First Term 3 and Common Ratio 2
In a geometric sequence starting with 3 and a common ratio of 2, the terms are 3, 6, 12, 24, and so on.
The third term is 3 2^(3-1) = 12.
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Sum of First n Natural Numbers
The sum of the first n natural numbers is given by the formula Sn = n(n+1)/2.
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Word Problem: Arithmetic Sequence in Savings
In a word problem, an arithmetic sequence might represent monthly savings that increase by a fixed amount, requiring you to find the total after n months.
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Word Problem: Geometric Sequence in Population Growth
In a word problem, a geometric sequence could model population growth where the population doubles each year, and you need to find the population after n years.
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Identifying the Type of Sequence
To identify the type of sequence, check if the difference between terms is constant for arithmetic or if the ratio is constant for geometric.
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Last Term of an Arithmetic Sequence
The last term of an arithmetic sequence can be found using an = a1 + (n-1)d, where n is the number of terms.
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Last Term of a Geometric Sequence
The last term of a geometric sequence is an = a1 r^(n-1), for the nth term.
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Finite vs. Infinite Sequences
A finite sequence has a limited number of terms, while an infinite sequence continues indefinitely.
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Partial Sum of a Sequence
A partial sum of a sequence is the sum of the first n terms, often denoted as Sn.
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Convergent Series
A convergent series is one where the sum approaches a finite limit as the number of terms increases, like an infinite geometric series with |r| < 1.
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Divergent Series
A divergent series is one where the sum does not approach a finite limit, such as an arithmetic series with a non-zero common difference.
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Recursive Sequence Example: Fibonacci
In the Fibonacci sequence, each term is the sum of the two before it, starting from 0 and 1.
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Arithmetic Series Trap: Including or Excluding the First Term
A common trap is forgetting whether the sum formula includes the first term correctly in calculations.
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Geometric Series Trap: When r = 1
If the common ratio is 1, the geometric series sum formula does not apply, as the terms do not converge.
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Sequence in Real-World Contexts
Sequences appear in real-world contexts like installment payments or compound interest, requiring pattern identification.
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Formula for the Sum of Even Numbers
The sum of the first n even numbers is n(n+1), as they form an arithmetic sequence starting from 2.
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Formula for the Sum of Squares
The sum of the first n squares is given by n(n+1)(2n+1)/6, which is a specific sequence pattern.
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Non-Linear Patterns in Sequences
Non-linear patterns in sequences might involve squares or cubes, like 1, 4, 9, 16, which are squares of natural numbers.
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Arithmetic Sequence with Negative Common Difference
An arithmetic sequence can have a negative common difference, making it decreasing, such as 10, 7, 4, 1.
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Geometric Sequence with Fraction Common Ratio
A geometric sequence might have a fractional common ratio, like 1, 1/2, 1/4, 1/8.
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Example: Sum of Arithmetic Sequence
For an arithmetic sequence 2, 4, 6, 8, the sum of the first 4 terms is 2 + 4 + 6 + 8 = 20.
Using the formula: S4 = 4/2 (2 + 8) = 20.