Geometric sequences
59 flashcards covering Geometric sequences for the SAT Math section.
A geometric sequence is a pattern of numbers where each term after the first is found by multiplying the previous term by a constant value, known as the common ratio. For instance, in the sequence 3, 6, 12, 24, each number is doubled to get the next one, making the common ratio 2. This concept helps model situations like exponential growth or decay, such as population increases or investment compounding, and it's a key part of algebra that builds on basic patterns.
On the SAT Math section, geometric sequences often appear in multiple-choice questions that require finding a specific term, calculating the sum of a series, or identifying the pattern in a word problem. Common traps include mistaking them for arithmetic sequences or errors in applying exponents, so watch for subtle wording that might hide the sequence type. Focus on memorizing and using formulas like the nth term (a_n = a_1 * r^(n-1)) and the sum of the first n terms (S_n = a_1 * (1 - r^n) / (1 - r)), and practice breaking down problems step by step.
Double-check the common ratio before proceeding.
Terms (59)
- 01
Geometric sequence
A sequence where each term after the first is obtained by multiplying the previous term by a constant called the common ratio.
- 02
Common ratio
The constant factor by which each term in a geometric sequence is multiplied to get the next term.
- 03
First term of a geometric sequence
The initial value in a geometric sequence, denoted as 'a', from which all subsequent terms are derived by multiplying by the common ratio.
- 04
Nth term of a geometric sequence
The term at position n in a geometric sequence, calculated using the formula a r^(n-1), where a is the first term and r is the common ratio.
- 05
Formula for the nth term
The expression a r^(n-1) that gives the nth term of a geometric sequence, where a is the first term, r is the common ratio, and n is the term number.
- 06
Sum of the first n terms
The total of the first n terms of a geometric sequence, calculated using the formula Sn = a (1 - r^n) / (1 - r) when r ≠ 1.
- 07
Geometric series
The sum of the terms of a geometric sequence, which can be finite or infinite depending on the number of terms.
- 08
Finite geometric series
A geometric series that includes only a specific number of terms, typically the first n terms of a geometric sequence.
- 09
Infinite geometric series
A geometric series that continues indefinitely, where the sum exists only if the absolute value of the common ratio is less than 1.
- 10
Condition for convergence
For an infinite geometric series, the series converges to a sum if the absolute value of the common ratio is less than 1; otherwise, it diverges.
- 11
Sum of an infinite geometric series
The value that an infinite geometric series approaches, given by the formula S = a / (1 - r) when the absolute value of r is less than 1.
- 12
Arithmetic vs. geometric sequence
An arithmetic sequence adds a constant difference to each term, while a geometric sequence multiplies each term by a constant ratio.
- 13
Identifying a geometric sequence
To determine if a sequence is geometric, check if the ratio between consecutive terms is constant.
- 14
Calculating the common ratio
Divide any term in a geometric sequence by the previous term to find the common ratio, ensuring consistency across the sequence.
- 15
Recursive definition of a geometric sequence
A way to define a geometric sequence where each term is expressed as the previous term multiplied by the common ratio, starting from the first term.
- 16
Explicit formula for a geometric sequence
A direct formula to find any term without listing prior terms, such as a r^(n-1) for the nth term.
- 17
Solving for the common ratio
Use the relationship between terms in a geometric sequence to set up an equation and solve for r, often by dividing two terms.
- 18
Finding the number of terms
Determine n in a geometric sequence by using the nth term formula and solving for n when given a specific term and the sequence parameters.
- 19
Sum when |r| > 1
In a geometric series, if the absolute value of the common ratio is greater than 1, the sum of the infinite series diverges and grows without bound.
- 20
Sum when |r| < 1
For an infinite geometric series, if the absolute value of the common ratio is less than 1, the series converges to a finite sum of a / (1 - r).
- 21
Applications in compound interest
Geometric sequences model compound interest, where the amount grows by a fixed ratio each period, such as annually.
- 22
Applications in exponential growth
Geometric sequences represent situations like population growth or radioactive decay, where quantities multiply by a constant factor over time.
- 23
Trap: Confusing multiplication and addition
A common error is treating a geometric sequence like an arithmetic one by adding instead of multiplying, leading to incorrect terms.
- 24
Trap: Incorrectly identifying the first term
Mistakenly selecting the wrong starting value can throw off the entire sequence, so always verify the initial term carefully.
- 25
Example: Sequence with r=2
In a geometric sequence starting with 3 and r=2, the terms are 3, 6, 12, 24, and so on.
For a=3 and r=2, the third term is 3 2^2 = 12.
- 26
Example: Sequence with r=1/2
A geometric sequence with first term 16 and r=1/2 has terms 16, 8, 4, 2, decreasing by half each time.
The fourth term is 16 (1/2)^3 = 2.
- 27
Example: Negative common ratio
In a geometric sequence with r=-3, terms alternate in sign and grow in magnitude, such as 2, -6, 18, -54.
Starting with 2, the third term is 2 (-3)^2 = 18.
- 28
Word problem: Bacteria growth
A scenario where bacteria double every hour, forming a geometric sequence to calculate population over time.
- 29
Word problem: Investment growth
An investment that increases by 5% annually follows a geometric sequence, used to find future values.
- 30
Deriving the sum formula
The formula for the sum of the first n terms is derived by manipulating the series as Sn - rSn = a(1 - r^n)/(1 - r) for r ≠ 1.
- 31
Partial sum formula
A formula that gives the sum of the first k terms of a geometric series, useful for calculating subtotals within a larger sequence.
- 32
Using sigma notation
Geometric series can be written in sigma notation, like sum from n=0 to infinity of ar^n, to represent the entire series compactly.
- 33
Geometric mean
In the context of sequences, it relates to the common ratio, but for two numbers, it's the square root of their product, often seen in sequence problems.
- 34
Ratio of terms
In a geometric sequence, the ratio between consecutive terms is constant, which is the key property distinguishing it from other sequences.
- 35
Sequences with fractions
Geometric sequences where terms are fractions, requiring careful handling of multiplication, such as 1/2, 1/4, 1/8.
- 36
Sequences with decimals
Geometric sequences involving decimal ratios, like 1.5, 2.25, 3.375, where each term is multiplied by a decimal common ratio.
- 37
Zero common ratio
If the common ratio is zero, the sequence becomes a, 0, 0, 0, and so on, which is technically geometric but often a special case.
- 38
All terms equal (r=1)
When the common ratio is 1, every term in the sequence is the same as the first term, resulting in a constant sequence.
- 39
Alternating sequences (r=-1)
A geometric sequence with r=-1 alternates between the first term and its negative, like 5, -5, 5, -5.
- 40
Finding missing terms
Use the common ratio to identify omitted terms in a geometric sequence by multiplying or dividing appropriately.
- 41
Verifying series convergence
Check if an infinite geometric series converges by ensuring the absolute value of the common ratio is less than 1.
- 42
Calculating sum for n terms
Plug values into the sum formula Sn = a (1 - r^n) / (1 - r) to find the total of the first n terms.
- 43
Error in formula application
A frequent mistake is forgetting to adjust for r=1, where the sum is simply an, or misplacing exponents in calculations.
- 44
Real-world example: Population decline
A population decreasing by 10% each year forms a geometric sequence, used to predict future sizes.
Starting with 1000, after two years it's 1000 0.9^2 = 810.
- 45
Real-world example: Sound intensity
Sound intensity in decibels can involve geometric sequences when dealing with multiples of a base intensity.
- 46
Strategy for solving sequence problems
First, identify if the sequence is geometric by checking ratios, then use the appropriate formula to find terms or sums.
- 47
Trap: Assuming r is positive
Sequences can have negative common ratios, leading to alternating signs, so always verify the sign in problems.
- 48
Fractional exponents in sequences
When the common ratio involves roots, like r=√2, the sequence grows or shrinks based on that factor.
- 49
Sum of even-numbered terms
For a geometric sequence, the sum of even-numbered terms can be found by treating it as a new sequence with adjusted first term and ratio.
- 50
Geometric sequence in patterns
Recognize patterns in figures or numbers that double or halve, indicating a geometric relationship.
- 51
Using logarithms in sequences
To solve for n when a term is given exponentially, take the logarithm of both sides in the nth term formula.
- 52
Trap: Dividing incorrectly for r
When calculating r, ensure you're dividing the correct terms in order to avoid getting the reciprocal.
- 53
Sequence with r=0
A geometric sequence where r=0 after the first term results in all zeros, affecting sums and patterns.
- 54
Partial fractions in sums
In some advanced problems, simplify sums of geometric series using partial fractions when combined with other expressions.
- 55
Geometric progression in algebra
A term for geometric sequences in algebraic contexts, emphasizing their progressive multiplication.
- 56
Inverse of a geometric sequence
The sequence formed by taking reciprocals, which may or may not be geometric depending on the original ratio.
- 57
Trap: Misinterpreting the index
In formulas, remember that n starts at 1 for sequences, so the first term is a r^0 = a.
- 58
Example: Decreasing sequence
A geometric sequence with r=0.8, starting at 100, gives terms 100, 80, 64, showing exponential decay.
The sum of the first three terms is 100 + 80 + 64 = 244.
- 59
Balanced geometric sequences
Sequences where the common ratio is a fraction that balances growth and decay in modeling scenarios.