Sequences and Series
Types of Sequence
● Progression → sequence or series
● Increasing sequence → each term greater than the previous term
● Decreasing sequence → each term less than the previous term
● Arithmetic sequence → difference between one term and the next is always
the same
● Geometric sequence → ratio of one term to the next is always the same
● Periodic sequence → repeats itself at regular intervals (number of terms
before repeat is called period)
● Series
○ The sum of all terms in a sequence
○ Use Σ notation
10
○ E.G ∑ ak means the series a1 + a2 + ... + a10
k=1
Arithmetic Progressions
● U n = a + (n − 1)d
○ U n is the term you desire to find
○ a is the first term
○ n is the term number you want to find
○ d is the common difference between the terms
n n
● Sn = 2 (2a + [n − 1]d) OR S n = 2 (f irst term + last term)
○ Formula for the sum to a certain number (ie the first n terms
summed together)
● This type of series diverges → S ∞ = ± ∞
Geometric Progressions
● U n = arn−1
○ U n is the term you desire to find
○ a is the first term
○ n is the term number you want to find
○ r is the common ratio between the terms
a(1−rn ) a(rn −1)
● Sn = 1−r OR S n = r−1
○ Formula for the sum to a certain number (ie the first n terms
summed together)
● If r is between 1 and -1, this type of series converges
a
● S∞ = 1−r f or − 1 < r < 1
Types of Sequence
● Progression → sequence or series
● Increasing sequence → each term greater than the previous term
● Decreasing sequence → each term less than the previous term
● Arithmetic sequence → difference between one term and the next is always
the same
● Geometric sequence → ratio of one term to the next is always the same
● Periodic sequence → repeats itself at regular intervals (number of terms
before repeat is called period)
● Series
○ The sum of all terms in a sequence
○ Use Σ notation
10
○ E.G ∑ ak means the series a1 + a2 + ... + a10
k=1
Arithmetic Progressions
● U n = a + (n − 1)d
○ U n is the term you desire to find
○ a is the first term
○ n is the term number you want to find
○ d is the common difference between the terms
n n
● Sn = 2 (2a + [n − 1]d) OR S n = 2 (f irst term + last term)
○ Formula for the sum to a certain number (ie the first n terms
summed together)
● This type of series diverges → S ∞ = ± ∞
Geometric Progressions
● U n = arn−1
○ U n is the term you desire to find
○ a is the first term
○ n is the term number you want to find
○ r is the common ratio between the terms
a(1−rn ) a(rn −1)
● Sn = 1−r OR S n = r−1
○ Formula for the sum to a certain number (ie the first n terms
summed together)
● If r is between 1 and -1, this type of series converges
a
● S∞ = 1−r f or − 1 < r < 1
