Sequences
49 flashcards covering Sequences for the SAT Math section.
Sequences are essentially ordered lists of numbers that follow a predictable pattern, making them a fundamental concept in math. For instance, a sequence might be 2, 4, 6, 8, where each number increases by 2—this is called an arithmetic sequence. Or it could be 3, 6, 12, 24, where each term multiplies by 2, known as a geometric sequence. Mastering sequences helps you spot patterns and solve real-world problems, and it's a key building block for more advanced topics like series and functions.
On the SAT Math section, sequences typically show up in multiple-choice questions that require finding the nth term, the sum of a sequence, or identifying the pattern in a given list. Common traps include mistaking one type of sequence for another or overlooking the starting term, which can lead to calculation errors. Focus on practicing formulas, such as the arithmetic sequence formula (a_n = a_1 + (n-1)d), and double-check patterns to ensure accuracy.
Remember to always identify the sequence type first.
Terms (49)
- 01
Sequence
A sequence is an ordered list of numbers that follows a specific pattern or rule, such as 2, 4, 6, 8.
- 02
Arithmetic sequence
An arithmetic sequence is a sequence where the difference between consecutive terms is constant, called the common difference.
- 03
Geometric sequence
A geometric sequence is a sequence where the ratio between consecutive terms is constant, called the common ratio.
- 04
Common difference
In an arithmetic sequence, the common difference is the fixed amount added to each term to get the next term, such as 3 in the sequence 5, 8, 11.
- 05
Common ratio
In a geometric sequence, the common ratio is the fixed number multiplied by each term to get the next term, such as 2 in the sequence 3, 6, 12.
- 06
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.
- 07
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.
- 08
Sum of an arithmetic series
The sum of the first n terms of an arithmetic series is given by Sn = n/2 (a1 + an), or Sn = n/2 [2a1 + (n-1)d], where a1 is the first term and d is the common difference.
- 09
Sum of a geometric series
The sum of the first n terms of a geometric series is given by Sn = a1 (1 - r^n) / (1 - r), where a1 is the first term, r is the common ratio, and r ≠ 1.
- 10
First term
In a sequence, the first term is the initial number in the list, which serves as the starting point for generating subsequent terms.
- 11
Last term
In a finite sequence or series, the last term is the final number in the list, often used in formulas for sums.
- 12
Number of terms
The number of terms in a sequence is the count of elements in the list, which can be determined from the pattern or formula.
- 13
Increasing sequence
An increasing sequence is one where each term is larger than the previous one, such as an arithmetic sequence with a positive common difference.
- 14
Decreasing sequence
A decreasing sequence is one where each term is smaller than the previous one, such as an arithmetic sequence with a negative common difference.
- 15
Constant sequence
A constant sequence is one where every term is the same, such as 7, 7, 7, which has a common difference of zero.
- 16
Finite sequence
A finite sequence has a limited number of terms, unlike an infinite sequence that continues indefinitely.
- 17
Infinite sequence
An infinite sequence continues forever according to its pattern, such as the natural numbers 1, 2, 3, and so on.
- 18
Recursive formula
A recursive formula defines each term of a sequence based on the previous term, such as an = a{n-1} + 2 for an arithmetic sequence.
- 19
Explicit formula
An explicit formula allows direct calculation of any term in a sequence without needing prior terms, like an = 3n + 1 for an arithmetic sequence.
- 20
Pattern recognition in sequences
Pattern recognition in sequences involves identifying the rule that generates the terms, such as adding a constant or multiplying by a factor.
- 21
Sigma notation
Sigma notation is a way to write the sum of a sequence, such as ∑ from i=1 to n of ai, which means adding the first n terms.
- 22
Arithmetic mean of a sequence
The arithmetic mean of a sequence is the average of its terms, calculated by summing them and dividing by the number of terms.
- 23
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 given first term and common difference or ratio.
- 24
Common trap: Confusing arithmetic and geometric sequences
A common error is mistaking an arithmetic sequence for a geometric one or vice versa, so always check if the pattern involves addition or multiplication.
- 25
Common trap: Off-by-one errors in indexing
In sequences, off-by-one errors occur when miscounting the term number, such as thinking the first term is n=1 but using n=0 in formulas.
- 26
Worked example: Next term in arithmetic sequence
For the arithmetic sequence 4, 7, 10, the next term is found by adding the common difference of 3, resulting in 13.
Sequence: 4, 7, 10; Next term: 13
- 27
Worked example: Nth term in geometric sequence
For the geometric sequence 2, 6, 18 with first term 2 and common ratio 3, the 4th term is 2 3^(4-1) = 54.
Sequence: 2, 6, 18; 4th term: 54
- 28
Sum of first few terms of arithmetic sequence
For an arithmetic sequence starting at 5 with common difference 2, the sum of the first 3 terms is 5 + 7 + 9 = 21.
- 29
Sum of first few terms of geometric sequence
For a geometric sequence starting at 3 with common ratio 2, the sum of the first 3 terms is 3 + 6 + 12 = 21.
- 30
Sequences in word problems
Sequences often appear in word problems involving patterns over time, like population growth or installment payments, requiring identification of the sequence type.
- 31
Reverse of a sequence
The reverse of a sequence is the same list of numbers in opposite order, which can help in understanding symmetry or solving certain problems.
- 32
Even and odd terms in a sequence
Even and odd terms refer to the positions, like the 2nd and 4th terms being even-positioned, which might be summed separately in problems.
- 33
Partial sum of a sequence
A partial sum is the total of the first k terms of a sequence, useful for calculating subtotals before the full sum.
- 34
Difference between sequence and series
A sequence is the list of terms, while a series is the sum of those terms, though they are related concepts in math problems.
- 35
Arithmetic sequence with zero common difference
An arithmetic sequence with a common difference of zero is a constant sequence, where all terms are identical.
- 36
Geometric sequence with ratio of 1
A geometric sequence with a common ratio of 1 has all terms equal to the first term, forming a constant sequence.
- 37
Geometric sequence with ratio greater than 1
In such a sequence, terms increase rapidly, like 1, 2, 4, 8, which grows exponentially.
- 38
Geometric sequence with ratio between 0 and 1
Terms decrease towards zero, such as 1, 0.5, 0.25, approaching but never reaching zero.
- 39
Strategy for summing sequences
To sum a sequence, determine if it's arithmetic or geometric, use the correct formula, and plug in the values for the first term, common difference or ratio, and number of terms.
- 40
Common trap: Dividing instead of multiplying in geometric sequences
Students might incorrectly divide terms instead of multiplying by the common ratio when extending a geometric sequence.
- 41
Worked example: Sum of arithmetic series
For the arithmetic sequence 2, 5, 8, 11 with 4 terms, the sum is 4/2 (2 + 11) = 2 13 = 26.
Sequence: 2, 5, 8, 11; Sum: 26
- 42
Worked example: Sum of geometric series
For the geometric sequence 1, 3, 9 with 3 terms and ratio 3, the sum is 1 (1 - 3^3) / (1 - 3) = 1 (1 - 27) / (-2) = 13.
Sequence: 1, 3, 9; Sum: 13
- 43
Sequences with fractions
Sequences can involve fractions, like an arithmetic sequence starting at 1/2 with common difference 1/4: 1/2, 3/4, 5/4.
- 44
Sequences with negative terms
Sequences may include negative numbers, such as an arithmetic sequence: -3, -1, 1, 3, with a common difference of 2.
- 45
Alternating sequences
An alternating sequence changes sign regularly, often seen in geometric sequences with a negative common ratio, like 2, -4, 8, -16.
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Finding the common difference from terms
To find the common difference, subtract one term from the next, such as in 10, 15, 20, where 15 - 10 = 5.
- 47
Finding the common ratio from terms
To find the common ratio, divide one term by the previous one, such as in 5, 10, 20, where 10 / 5 = 2.
- 48
Error in assuming linear growth
A common mistake is assuming all sequences grow linearly, when some, like geometric ones, grow exponentially.
- 49
Using sequences for patterns in data
Sequences help model real-world patterns, such as the number of bacteria doubling each hour in exponential growth.