Geometric sequences
64 flashcards covering Geometric sequences for the GMAT Quantitative section.
Geometric sequences are lists of numbers where each term after the first is obtained by multiplying the previous term by a constant value, called the common ratio. For example, in the sequence 5, 10, 20, 40, each number is doubled to get the next one. This pattern is useful for modeling real-world scenarios like exponential growth in investments or population expansion, making it a key concept in algebra and beyond.
On the GMAT Quantitative section, geometric sequences often appear in problem-solving and data sufficiency questions, where you might need to find a specific term, calculate the sum of a series, or identify patterns in word problems. Common traps include mistaking them for arithmetic sequences or errors in handling fractions and negative ratios, so double-check your common ratio. Focus on memorizing formulas like the nth term (a * r^(n-1)) and the sum of the first n terms, as these help solve problems efficiently.
Practice verifying the common ratio before proceeding.
Terms (64)
- 01
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 ratio
The common ratio in a geometric sequence is the fixed number that multiplies any term to get the next term.
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First term of a geometric sequence
The first term, often denoted as a, is the initial number in a geometric sequence from which all subsequent terms are derived by multiplying by the common ratio.
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nth term formula
The formula for the nth term of a geometric sequence is an = a r^(n-1), where a 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
The sum of the first n terms of a geometric sequence is given by Sn = a (1 - r^n) / (1 - r), provided r is not equal to 1.
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Infinite geometric series
An infinite geometric series is the sum of an endless geometric sequence, which converges to a finite value only if the absolute value of the common ratio is less than 1.
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Convergence of infinite series
An infinite geometric series converges if the absolute value of the common ratio is less than 1, meaning the terms get progressively smaller and approach zero.
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Sum of infinite geometric series
The sum of an infinite geometric series is S = a / (1 - r), where a is the first term and r is the common ratio, but only if |r| < 1.
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Identifying a geometric sequence
To identify a geometric sequence, check if the ratio between consecutive terms is constant; if it is, the sequence is geometric.
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Calculating the common ratio
To calculate the common ratio, divide any term in the sequence by the previous term; this value should be the same for all pairs of consecutive terms.
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Geometric sequence with r greater than 1
In a geometric sequence where the common ratio r is greater than 1, the terms increase rapidly, making the sequence diverge if summed infinitely.
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Geometric sequence with 0 < r < 1
In a geometric sequence where the common ratio r is between 0 and 1, the terms decrease and approach zero, allowing an infinite series to converge.
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Geometric sequence with negative r
In a geometric sequence with a negative common ratio, the terms alternate in sign, which can affect the sum and behavior of the series.
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Sum when |r| < 1
When the absolute value of the common ratio is less than 1, the infinite geometric series sums to a finite value using the formula S = a / (1 - r).
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Sum when |r| > 1
When the absolute value of the common ratio is greater than 1, the infinite geometric series diverges and does not sum to a finite value.
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Arithmetic vs. geometric sequence
An arithmetic sequence adds a constant difference to each term, while a geometric sequence multiplies by a constant ratio; confusing them is a common error on tests.
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Recursive formula for geometric sequence
The recursive formula for a geometric sequence defines each term as the previous term multiplied by the common ratio, such as an = a{n-1} r.
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Explicit formula for geometric sequence
The explicit formula directly gives the nth term as a r^(n-1), allowing you to find any term without calculating the previous ones.
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Finding the nth term example
To find the nth term, use the formula an = a r^(n-1); for instance, in the sequence 3, 6, 12, with a=3 and r=2, the 4th term is 3 2^3 = 24.
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Sum of first n terms example
To calculate the sum of the first n terms, use Sn = a (1 - r^n) / (1 - r); for example, for 2, 4, 8 with n=3, S3 = 2 (1 - 2^3) / (1 - 2) = 2 (1 - 8) / (-1) = 14.
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Infinite series sum example
For an infinite geometric series like 1 + 1/2 + 1/4 + ..., with a=1 and r=0.5, the sum is 1 / (1 - 0.5) = 2, since |r| < 1.
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Strategy for checking ratios
When solving sequence problems, always verify the ratio between terms to confirm it's geometric, as this prevents misapplication of formulas.
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Geometric mean
The geometric mean of two numbers is the square root of their product, which relates to geometric sequences as the middle term in a two-term sequence.
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Applications in population growth
Geometric sequences model population growth where each period multiplies the population by a constant factor, such as in exponential increase scenarios.
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Applications in compound interest
Compound interest uses a geometric sequence to calculate future values, where each period's amount is the previous amount multiplied by (1 + interest rate).
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Trap: Assuming constant difference
A common trap is assuming a sequence is arithmetic when it's geometric, so always check for multiplication rather than addition between terms.
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Partial sum of geometric series
The partial sum is the total of the first n terms of a geometric series, calculated using Sn = a (1 - r^n) / (1 - r), useful for finite approximations.
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Doubling time in sequences
Doubling time in a geometric sequence is the number of periods needed for a value to double, calculated using the formula for when a r^n = 2a, so r^n = 2.
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Sequence with r = 1
If the common ratio is 1, the sequence is constant, and the sum of the first n terms is simply n times the first term, not using the standard geometric formula.
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Sequence with r = 0
If the common ratio is 0, the sequence becomes a, 0, 0, 0, ..., and the sum of the first n terms is just a, as all subsequent terms are zero.
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Finding n for a given term
To find n when a specific term is given, solve a r^(n-1) = target value for n, which often requires logarithms if r is not 1.
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Logarithms in geometric sequences
Logarithms are used in geometric sequences to solve for the exponent, such as in finding how many terms it takes to reach a certain value.
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Geometric sequence in decay
Geometric sequences with 0 < r < 1 model decay, like radioactive half-life, where each term is a fraction of the previous one.
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Sum formula derivation
The sum formula for a geometric series is derived by writing out the terms and factoring, such as Sn = a + a r + a r^2 + ... + a r^(n-1), then multiplying by (1 - r).
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Error in infinite sum calculation
A frequent error is calculating the infinite sum when |r| >= 1, which doesn't converge, leading to incorrect answers in problems.
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Fractional common ratio
A fractional common ratio, like 1/2, results in a decreasing sequence that converges when summed infinitely, common in probability problems.
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Geometric sequence patterns
Patterns in geometric sequences include exponential growth or decay, recognizable by the powers of the common ratio in each term.
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Word problem: Investment growth
In word problems, geometric sequences describe investment growth, such as an initial amount doubling each year, requiring the sum formula for total value.
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Strategy for series vs. sequence
Distinguish between a sequence (list of terms) and a series (sum of terms) to avoid confusion when problems ask for sums rather than individual terms.
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Negative terms in sequence
Geometric sequences can have negative terms if the common ratio is negative, leading to alternating signs that affect the overall sum.
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Sum of alternating series
For an alternating geometric series with |r| < 1, the infinite sum is a / (1 - r), which converges despite the sign changes.
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Trap: Rounding errors
In calculations with geometric sequences, rounding intermediate steps can lead to errors, so maintain precision until the final answer.
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Geometric mean in sequences
In a geometric sequence, the geometric mean of two terms is the term in between them, useful for interpolation in problems.
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Reverse engineering a sequence
To reverse engineer, given terms, find a and r by solving equations from the sequence definition.
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Example: 5, 10, 20, 40
This is a geometric sequence with first term 5 and common ratio 2, where each term doubles the previous one.
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Non-integer common ratio
A common ratio that is not an integer, like 1.5, still forms a valid geometric sequence, often appearing in real-world applications.
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Sum for n approaching infinity
As n approaches infinity in a geometric series with |r| < 1, the sum approaches the infinite sum formula, a key concept for limits.
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Strategy: Factor out first term
When working with sums, factor out the first term to simplify the formula, making calculations easier and less error-prone.
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Geometric sequence in geometry
Geometric sequences can relate to geometric shapes, like side lengths in similar figures that scale by a constant ratio.
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Trap: Misordering terms
Ensure terms are in the correct order when applying formulas, as reversing them can lead to incorrect common ratios.
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Calculating partial sums
Partial sums help approximate the total of a series, especially when n is large, by using the finite sum formula.
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Example: Sum of 1 + 3 + 9 + 27
For the sequence 1, 3, 9, 27 with a=1 and r=3, the sum of the first 4 terms is 1 (1 - 3^4) / (1 - 3) = 1 (1 - 81) / (-2) = 40.
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Common ratio as fraction
When the common ratio is a fraction, like 2/3, the sequence decreases, and careful calculation is needed for sums.
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Infinite series with r=0.8
For an infinite series starting with 10 and r=0.8, the sum is 10 / (1 - 0.8) = 50, illustrating convergence.
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Strategy for word problems
In word problems involving geometric sequences, identify the initial value and the multiplier to set up the correct formula.
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Diverging sequence behavior
A geometric sequence with |r| > 1 diverges, meaning terms grow without bound, which is important for understanding limits.
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Example: 4, 2, 1, 0.5
This geometric sequence has a=4 and r=0.5, showing a decreasing pattern that converges when summed infinitely to 4 / (1 - 0.5) = 8.
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nth root in sequences
The nth root can relate to common ratios in problems, such as finding r when terms are raised to powers.
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Trap: Forgetting to subtract 1 in formula
In the nth term formula, remember to use n-1 as the exponent; forgetting this leads to off-by-one errors.
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Geometric progression in patterns
Geometric progressions are patterns where each element is multiplied by a constant, often tested in pattern recognition questions.
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Sum inequality for r > 1
For r > 1, the partial sums increase without limit, so no finite sum exists for the infinite series.
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Example: Finding r from terms
Given terms 16 and 64 in a geometric sequence, r = 64 / 16 = 4, allowing reconstruction of the full sequence.
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Advanced: Logarithmic solving
For complex problems, use logarithms to solve for n in equations like a r^n = value, such as n = log(value / a) / log(r).
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Balanced coverage in study
Ensure study includes both finite and infinite aspects, as GMAT tests a range of applications for geometric sequences.