Arithmetic sequences
52 flashcards covering Arithmetic sequences for the GMAT Quantitative section.
Arithmetic sequences are a type of number pattern where each term increases or decreases by a constant amount, called the common difference. For instance, the sequence 3, 7, 11, 15 has a common difference of 4, making it easy to predict the next terms or find the sum of the series. This concept is a building block in algebra and is useful for analyzing real-world scenarios like financial growth or scheduling, helping you develop logical problem-solving skills that are tested in math.
On the GMAT Quantitative section, arithmetic sequences typically show up in data sufficiency and problem-solving questions, where you might need to calculate the nth term, sum of terms, or identify patterns in sets of numbers. Common traps include mistaking them for geometric sequences or overlooking negative differences, which can lead to calculation errors. Focus on understanding key formulas, like the nth term formula (a_n = a_1 + (n-1)d), and practicing with word problems to apply them accurately under time pressure.
A concrete tip: Always double-check the common difference to avoid missteps.
Terms (52)
- 01
Arithmetic sequence
An arithmetic sequence is a list of numbers where the difference between consecutive terms is constant.
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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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First term of an arithmetic sequence
The first term is the initial number in an arithmetic sequence, from which subsequent terms are generated by adding the common difference.
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nth term of an arithmetic sequence
The nth term is any specific term in the sequence, found by starting with the first term and adding the common difference (n-1) times.
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Formula for the nth term
The formula for the nth term of an arithmetic sequence is an = a1 + (n-1)d, where an is the nth term, a1 is the first term, n is the term number, and d is the common difference.
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Arithmetic series
An arithmetic series is the sum of the terms in an arithmetic sequence.
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Sum of the first n terms
The sum of the first n terms of an arithmetic sequence is the total when you add up the initial n numbers in the sequence.
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Formula for the sum of an arithmetic series
The formula for the sum of the first n terms of an arithmetic sequence is Sn = n/2 (a1 + an), or Sn = n/2 [2a1 + (n-1)d], where Sn is the sum, a1 is the first term, an is the nth term, n is the number of terms, and d is the common difference.
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How to find the number of terms
To find the number of terms in an arithmetic sequence, use the formula n = [(last term - first term) / common difference] + 1, ensuring the result is a whole number.
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Increasing arithmetic sequence
An increasing arithmetic sequence has a positive common difference, so each term is larger than the previous one.
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Decreasing arithmetic sequence
A decreasing arithmetic sequence has a negative common difference, so each term is smaller than the previous one.
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Constant arithmetic sequence
A constant arithmetic sequence has a common difference of zero, meaning all terms are the same.
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Verifying if a sequence is arithmetic
To verify if a sequence is arithmetic, check if the difference between each pair of consecutive terms is the same.
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Difference between arithmetic and geometric sequences
An arithmetic sequence has a constant difference between terms, while a geometric sequence has a constant ratio between terms, which is a common trap on the GMAT.
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Common trap: Assuming a sequence is arithmetic
A common trap is assuming a sequence is arithmetic without verifying the constant difference, which can lead to errors in calculations.
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Example of an arithmetic sequence with positive difference
For example, the sequence 3, 7, 11, 15 has a common difference of 4.
In this sequence, adding 4 each time gives the next term.
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Example of an arithmetic sequence with negative difference
For example, the sequence 10, 7, 4, 1 has a common difference of -3.
Subtracting 3 from each term produces the next one.
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Sum of an even number of terms
When summing an even number of terms in an arithmetic sequence, the formula works the same, but the average of the first and last terms is the midpoint.
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Sum of an odd number of terms
In an arithmetic sequence with an odd number of terms, the middle term is the average of all terms.
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Midpoint of an arithmetic sequence
The midpoint of an arithmetic sequence is the average of the first and last terms, which equals the average of all terms in the sequence.
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Average of terms in an arithmetic sequence
The average of the terms in an arithmetic sequence is the same as the average of the first and last terms.
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Recursive formula for arithmetic sequence
The recursive formula for an arithmetic sequence defines each term as the previous term plus the common difference, such as an = a{n-1} + d.
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Explicit formula for arithmetic sequence
The explicit formula directly gives the nth term without needing prior terms, using an = a1 + (n-1)d.
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Solving for common difference
To solve for the common difference, subtract the first term from the second term, or use two terms and the formula.
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Solving for first term
To solve for the first term, rearrange the nth term formula or use given terms and the common difference.
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Word problem: Arithmetic sequence in salaries
In word problems, an arithmetic sequence might represent annual salary increases, where each year's salary adds a fixed amount to the previous year.
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Word problem: Installment payments
An arithmetic sequence can model installment payments that decrease by a fixed amount each period, such as loan repayments.
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Common trap: Forgetting to include the first term
A common trap is omitting the first term when calculating the sum, which underestimates the total.
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Common trap: Miscalculating the last term
Miscalculating the last term by not applying the common difference correctly can lead to errors in sums or sequences.
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Strategy: Use the formula directly
A strategy for GMAT problems is to plug values into the nth term or sum formula immediately to avoid manual listing of terms.
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Strategy: Work backwards from the sum
To solve for unknowns, work backwards using the sum formula, such as finding the first term if the sum and other details are given.
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Properties: Terms are equally spaced
In an arithmetic sequence, the terms are equally spaced on a number line, with the distance equal to the common difference.
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Properties: Difference is constant
The key property is that the difference between any two successive terms remains constant throughout the sequence.
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Finding the 10th term
To find the 10th term, use the formula a10 = a1 + 9d, where a1 is the first term and d is the common difference.
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Finding the sum of first 5 terms
To find the sum of the first 5 terms, use S5 = 5/2 (a1 + a5), or S5 = 5/2 [2a1 + 4d].
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Arithmetic sequence with fractions
An arithmetic sequence can include fractions, such as 1/2, 3/4, 1, where the common difference is 1/4.
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Arithmetic sequence with decimals
An arithmetic sequence might involve decimals, like 0.5, 1.0, 1.5, with a common difference of 0.5.
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Negative terms in sequence
An arithmetic sequence can have negative terms, such as -2, 0, 2, where the common difference is 2.
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Zero as a term
Zero can be a term in an arithmetic sequence, for example, in -1, 0, 1, with a common difference of 1.
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All positive terms
An arithmetic sequence with all positive terms occurs when the first term is positive and the common difference is positive or zero.
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All negative terms
An arithmetic sequence with all negative terms happens when the first term is negative and the common difference is zero or negative but keeps terms negative.
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Mixed signs in sequence
An arithmetic sequence can have mixed positive and negative terms, like -1, 0, 1, 2, depending on the first term and common difference.
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Largest term in a sequence
In an increasing arithmetic sequence, the largest term is the last one; in a decreasing one, it is the first.
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Smallest term in a sequence
In an increasing arithmetic sequence, the smallest term is the first; in a decreasing one, it is the last.
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Arithmetic sequence of odd numbers
The sequence of odd numbers, like 1, 3, 5, is an arithmetic sequence with a common difference of 2.
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Arithmetic sequence of even numbers
The sequence of even numbers, such as 2, 4, 6, is an arithmetic sequence with a common difference of 2.
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Arithmetic sequence of multiples
A sequence of multiples, like 5, 10, 15, is an arithmetic sequence with a common difference equal to the multiple's base, here 5.
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Finite arithmetic sequence
A finite arithmetic sequence has a limited number of terms, unlike an infinite one that continues indefinitely.
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Infinite arithmetic sequence
An infinite arithmetic sequence theoretically goes on forever, but GMAT problems typically deal with finite portions.
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Using sequences in inequalities
In GMAT problems, arithmetic sequences can be used in inequalities to compare sums or terms, such as determining when a sum exceeds a value.
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Common trap: Incorrect order of terms
A common trap is reversing the order of terms when calculating sums, which alters the result.
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Strategy: Pair terms for summing
A strategy is to pair the first and last terms, second and second-last, etc., as each pair sums to the same value in an arithmetic sequence.