Sequences and series
59 flashcards covering Sequences and series for the GMAT Quantitative section.
Sequences and series are patterns of numbers that follow specific rules, making them essential tools in math for understanding growth and summation. A sequence is simply an ordered list of numbers, like 1, 3, 5, 7, where each term is generated by a rule. A series takes this a step further by adding up the terms in a sequence, such as summing those numbers to find a total. These concepts help model real-world scenarios, like population growth or financial investments, and build a foundation for more advanced math.
On the GMAT Quantitative section, sequences and series appear in problem-solving and data sufficiency questions, often involving arithmetic or geometric patterns. You'll need to identify the nth term, calculate sums, or detect errors in given sequences. Common traps include mistaking one type of sequence for another or overlooking alternating signs, so focus on memorizing key formulas like the sum of an arithmetic series and practicing quick pattern recognition to save time. For better results, always jot down the first few terms to clarify the pattern.
Terms (59)
- 01
Sequence
A sequence is an ordered list of numbers following a specific pattern or rule.
- 02
Series
A series is the sum of the terms in a sequence.
- 03
Arithmetic Sequence
An arithmetic sequence is a sequence where the difference between consecutive terms is constant.
- 04
Common Difference
The common difference in an arithmetic sequence is the fixed amount added to each term to get the next one.
- 05
nth Term of Arithmetic Sequence
The nth term of an arithmetic sequence is calculated using the formula an = a1 + (n-1)d, where a1 is the first term, d is the common difference, and n is the term number.
- 06
Sum of Arithmetic Series
The sum of the first n terms of an arithmetic series is given by Sn = n/2 (a1 + an), where a1 is the first term and an is the nth term.
- 07
Geometric Sequence
A geometric sequence is a sequence where each term after the first is found by multiplying the previous one by a constant ratio.
- 08
Common Ratio
The common ratio in a geometric sequence is the fixed number by which each term is multiplied to get the next term.
- 09
nth Term of Geometric Sequence
The nth term of a geometric sequence is given by an = a1 r^(n-1), where a1 is the first term, r is the common ratio, and n is the term number.
- 10
Sum of Finite Geometric Series
The sum of the first n terms of a geometric series is Sn = a1 (1 - r^n) / (1 - r), where a1 is the first term and r is the common ratio, provided r ≠ 1.
- 11
Sum of Infinite Geometric Series
The sum of an infinite geometric series is S = a1 / (1 - r), where |r| < 1, meaning the series converges.
- 12
Convergent Series
A convergent series is one where the sum approaches a finite limit as the number of terms increases indefinitely.
- 13
Divergent Series
A divergent series is one where the sum does not approach a finite limit and grows without bound.
- 14
First Term of a Sequence
The first term of a sequence is the initial value from which subsequent terms are generated based on the pattern.
- 15
Last Term in a Sequence
The last term in a finite sequence is the final value, which can be found using the nth term formula for arithmetic or geometric sequences.
- 16
Number of Terms in a Sequence
The number of terms in a sequence can be determined by solving for n in the nth term formula when the last term is known.
- 17
Recursive Definition of Sequence
A recursive definition of a sequence defines each term based on the previous term, such as an = a{n-1} + d for an arithmetic sequence.
- 18
Explicit Formula for Sequence
An explicit formula for a sequence allows direct calculation of any term without knowing prior terms, like an = a1 + (n-1)d for arithmetic sequences.
- 19
Sigma Notation
Sigma notation is a compact way to write the sum of a series, using the symbol Σ to indicate addition from a starting index to an ending index.
- 20
Partial Sum of a Series
A partial sum of a series is the sum of the first n terms, which helps in understanding how the total sum builds up.
- 21
Finite Series
A finite series has a limited number of terms, and its sum can be calculated exactly using appropriate formulas.
- 22
Infinite Series
An infinite series has an unlimited number of terms, and it converges only if the sum approaches a specific value.
- 23
Convergence Condition
For a geometric series, convergence occurs when the absolute value of the common ratio is less than 1.
- 24
Pattern Recognition in Sequences
Pattern recognition in sequences involves identifying the rule, such as addition or multiplication, that relates consecutive terms.
- 25
Arithmetic Mean in Sequences
In an arithmetic sequence, the arithmetic mean of the terms is the average of the first and last terms for any subset.
- 26
Strategy for Summing Series
To sum a series efficiently, identify whether it is arithmetic or geometric and apply the corresponding formula to avoid manual addition.
- 27
Common Trap: Miscounting Terms
A common error is miscounting the number of terms in a sequence, often by forgetting to adjust for the starting index.
- 28
Common Trap: Incorrect Ratio
In geometric sequences, assuming the wrong common ratio can lead to errors in calculating subsequent terms or sums.
- 29
Word Problem: Arithmetic Growth
In word problems involving arithmetic growth, such as population increase by a fixed amount each year, use the arithmetic sequence sum formula.
- 30
Word Problem: Geometric Decay
Geometric decay problems, like depreciation of an asset by a fixed percentage annually, require the geometric series sum formula.
- 31
Applications in Finance
Sequences and series are used in finance for concepts like annuities, where payments form a geometric series.
- 32
Population Growth Model
A population growth model often uses a geometric sequence if growth is proportional, such as doubling every period.
- 33
Compound Interest Formula
The compound interest formula involves a geometric sequence, where the amount grows by a factor each period.
- 34
Depreciation of Assets
Asset depreciation can be modeled as an arithmetic sequence if it decreases by a fixed amount each year.
- 35
Inflation Adjustment
Inflation adjustment over time can involve a geometric series if rates compound annually.
- 36
Summing Consecutive Integers
The sum of the first n consecutive integers is an arithmetic series with formula Sn = n(n+1)/2.
- 37
Sum of Even Numbers
The sum of the first n even numbers is an arithmetic series with first term 2 and common difference 2.
- 38
Sum of Odd Numbers
The sum of the first n odd numbers is an arithmetic series that equals n squared.
- 39
Difference Between Sequence and Series
The key difference is that a sequence is a list of numbers, while a series is the result of adding those numbers together.
- 40
Bounded Sequence
A bounded sequence is one where all terms are confined within a fixed range, either above or below certain values.
- 41
Monotonic Sequence
A monotonic sequence is one that is either entirely non-increasing or non-decreasing throughout its terms.
- 42
Limit of a Sequence
The limit of a sequence is the value that the terms approach as n approaches infinity, if it exists.
- 43
Arithmetic Progression
Arithmetic progression is another term for an arithmetic sequence, emphasizing the progressive addition of a constant.
- 44
Geometric Progression
Geometric progression refers to a geometric sequence, highlighting the multiplicative progression.
- 45
Error in Assuming Constant Difference
A frequent mistake is assuming a sequence has a constant difference when it actually follows a different pattern, like geometric.
- 46
Finding the Sum Using Formulas
To find the sum of a series, always verify the type and use the precise formula to avoid calculation errors.
- 47
Telescoping Series
A telescoping series is one where most terms cancel out when summed, simplifying the calculation.
- 48
Alternating Series
An alternating series has terms that alternate in sign, and for geometric ones, convergence depends on the ratio.
- 49
Binomial Series
The binomial series expands expressions like (1 + x)^n, which can be treated as an infinite geometric series under certain conditions.
- 50
Partial Fractions in Series
In some series problems, partial fractions help break down rational expressions for easier summing.
- 51
Ratio Test for Convergence
The ratio test checks if a series converges by comparing the limit of the ratio of consecutive terms to 1.
- 52
Harmonic Sequence
A harmonic sequence is one where the reciprocals of the terms form an arithmetic sequence.
- 53
Average of Terms in Arithmetic Sequence
The average of all terms in an arithmetic sequence equals the average of the first and last terms.
- 54
Geometric Sequence Ratio Calculation
To calculate the common ratio, divide any term by the previous term in a geometric sequence.
- 55
Series in Probability
In probability, infinite series like geometric distributions sum to calculate total probabilities.
- 56
Deriving Arithmetic Sum Formula
The arithmetic sum formula can be derived by pairing terms from the beginning and end of the sequence.
- 57
Deriving Geometric Sum Formula
The geometric sum formula is derived by multiplying the series by the common ratio and subtracting.
- 58
Off-by-One Error in Sequences
An off-by-one error occurs when indexing starts incorrectly, such as confusing n=1 with n=0.
- 59
Using Sequences in Data Patterns
Sequences help identify patterns in data sets, such as trends in GMAT quantitative problems.