Sequences and series (Definitions and formulas)
Converging
"Means getting closer and closer to a sum, or limit"
| r | < 1
A series converges only if ________.
sⁿ = n/2 (a1 +an)
Formula to find the sum of an arithmetic Sequence
Explicit formula
A formula to find the nth term of a Sequence
Recursive formula
A formula to find the nth term of a sequence that uses previous term(s)
d
A notation to signify the common difference between two consecutive terms in an Arithmetic Sequence
Term
A number in a sequence
Geometric sequence
A sequence of numbers with a common "ratio" or multiplier between any two consecutive terms
Arithmetic sequence
A sequence of numbers with a common difference between any two consecutive terms
Geometric series
A sum of the terms in a geometric sequence
Sequence
An ordered list of numbers
Infinite sequence
An ordered list of numbers that continues forever
Finite sequence
An ordered list of numbers with a definite stopping point
d = an - a(n-1) .
Formula to find the common difference
r = an / a(n-1) .
Formula to find the common ratio
an = a₁ (rⁿ⁻¹)
Formula to find the nth term in a geometric series
an = a₁ + d(n-1)
Formula to find the nth term in an arithmetic Sequence
sⁿ = n/2 [ a1+ (a1 + (n-1)d) ]
Formula to find the sum of a finite arithmetic Sequence if the last term is unknown
sⁿ = a₁ (1 - rⁿ) / (1-r)
Formula to find the sum of a geometric series
s = a₁ / (1-r)
Formula to find the sum of an Infinite Geometric Series
Take any term and subtract the term before it.
How do you find the common difference?
Take any term and divide by the term before it.
How do you find the common ratio?
Common difference
In an arithmetic sequence it is the fixed number added to each term to equal the next consecutive term or the difference between two consecutive terms.
index
The "place" of the term in a sequence of numbers, it is the subscript used in formulas
Series
The "sum" of an ordered list of terms
Common ratio
The constant ratio or multiplier used in a geometric Sequence
sn
The notation to signify the "sum" of an ordered list of terms (or a sequence) in a finite arithmetic or geometric series
r
The notation to signify the common ratio between two consecutive terms in a geometric sequence
a1 or t1
The notation to signify the first term in a sequence
an or tn
The notation to signify the nth term in a sequence
n
The notation to signify the number of terms in a Sequence
Infinite geometric series
The sum of a an infinite sequence of numbers separated by a common ratio.
Infinite arithmetic series
The sum of an infinite sequence of numbers with a common difference
Sum of an arithmetic series
The sum of the first and last terms times half the number of terms?
Arithmetic series
The sum of the terms in an arithmetic sequence
Converges
The the sum of an finite geometric series can be found only if the series _________.