Geometric sequences are mathematical patterns where each term is a constant multiple of the previous one. This constant multiplier, called the , defines the sequence's behavior and growth rate.
Understanding geometric sequences is crucial for modeling exponential growth and decay in various fields. These sequences form the foundation for more complex mathematical concepts like series, limits, and financial calculations.
Geometric Sequences
Common ratio calculation
Common ratio r constant factor each term multiplied by to get next term
For a1,a2,a3,..., common ratio calculated as r=anan+1 for any n≥1
Sequence 2,6,18,54,..., common ratio r=26=618=1854=3
Positive common ratio sequence terms remain positive
Negative common ratio sequence alternates between positive and negative values (1, -2, 4, -8, ...)
Absolute value of common ratio greater than 1 sequence grows exponentially (2, 6, 18, 54, ...)
Absolute value of common ratio less than 1 sequence converges to 0 (1, 0.5, 0.25, 0.125, ...)
When the absolute value of the common ratio is less than 1, the sequence has a limit of 0 as n approaches infinity (infinite geometric sequence)
Term generation for geometric sequences
Generate terms of geometric sequence need first term a1 and common ratio r
Each subsequent term found by multiplying previous term by common ratio
Second term a2=a1⋅r
Third term a3=a2⋅r=a1⋅r2
Fourth term a4=a3⋅r=a1⋅r3
n-th term of geometric sequence given by an=a1⋅rn−1
First term a1=2 and common ratio r=3, fourth term a4=2⋅34−1=2⋅33=2⋅27=54
A geometric sequence with a fixed number of terms is called a finite geometric sequence
Recursive formulas in sequence analysis
for geometric sequence defines each term based on previous term
Recursive formula for geometric sequence an=an−1⋅r for n≥2
Each term equal to previous term multiplied by common ratio
To use recursive formula, need first term a1 and common ratio r
a1=3 and r=2, recursive formula an=an−1⋅2 for n≥2
Explicit formula useful for finding specific terms without calculating entire sequence (2, 6, 18, 54, 162, ...)
Explicit formula also used to solve problems involving sum of geometric series, where Sn=1−ra1(1−rn) for r=1
Sum of first 5 terms of sequence with a1=2 and r=3: S5=1−32(1−35)=−22(1−243)=242
The sum of a finite number of terms in a geometric sequence is called a partial sum
Related Sequences
Geometric sequences are closely related to arithmetic sequences
In an arithmetic sequence, the difference between consecutive terms is constant
While geometric sequences have a constant ratio, arithmetic sequences have a constant difference between terms
Key Terms to Review (3)
Common ratio: In a geometric sequence, the common ratio is the constant factor between consecutive terms. It is usually denoted by $r$ and can be found by dividing any term by its preceding term.
Geometric sequence: A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. It can be expressed as $a, ar, ar^2, ar^3, \ldots$ where $a$ is the first term and $r$ is the common ratio.
Recursive formula: A recursive formula defines each term of a sequence using the preceding term(s). It typically includes an initial condition and a recurrence relation.