Geometric Sequences and Series Essentials
Master key formulas, concepts, and convergence rules for geometric sequences and series.
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Questions Covered in This Set
8 cards to master
What is the formula for the nth term of a geometric sequence?
aₙ = a · r^(n-1), where a is the first term and r is the common ratio
What is the formula for the sum of the first n terms of a geometric series?
Sₙ = a · (1 - rⁿ)/(1 - r) when r ≠ 1
When does an infinite geometric series converge?
When |r| < 1, meaning the absolute value of the common ratio is less than 1
What is the formula for the sum of an infinite geometric series?
S = a/(1 - r) when |r| < 1
What is a common ratio in a geometric sequence?
The constant value by which each term is multiplied to get the next term
What happens to an infinite geometric series when |r| ≥ 1?
The series diverges, meaning it grows without bound or oscillates and has no finite sum
In the sequence 5, 15, 45, 135, what is the common ratio?
r = 3 (each term is multiplied by 3 to get the next term)
What is the sum of the infinite series 1/2 + 1/4 + 1/8 + 1/16 + ...?
S = 1, calculated using S = (1/2)/(1 - 1/2) = 1
