Calc2: 8.1-8.2 Sequences and series (Definitions and formulas)

Sequence

An ordered list of numbers

Series

The "sum" of an ordered list of terms

Arithmetic sequence

A sequence of numbers with a common difference between any two consecutive terms

Arithmetic series

The sum of the terms in an arithmetic sequence

Geometric sequence

A sequence of numbers with a common "ratio" or multiplier between any two consecutive terms

Geometric series

A sum of the terms in a geometric sequence

Finite sequence

An ordered list of numbers with a definite stopping point

Infinite sequence

An ordered list of numbers that continues forever

Infinite arithmetic series

The sum of an infinite sequence of numbers with a common difference

Infinite geometric series

The sum of a an infinite sequence of numbers separated by a 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.

Common ratio

The constant ratio or multiplier used in a geometric 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)

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?

index

The "place" of the term in a sequence of numbers, it is the subscript used in formulas

Term

A number in a sequence

n

The notation to signify the number of terms in a Sequence

r

The notation to signify the common ratio between two consecutive terms in a geometric sequence

d

A notation to signify the common difference between two consecutive terms in an Arithmetic Sequence

sn

The notation to signify the "sum" of an ordered list of terms (or a sequence) in a finite arithmetic or geometric series

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

Converges

The the sum of an finite geometric series can be found only if the series _________.

Converging

"Means getting closer and closer to a sum, or limit"

Sum of an arithmetic series

The sum of the first and last terms times half the number of terms?

| r | < 1

A series converges only if ________.

an = a₁ + d(n-1)

Formula to find the nth term in an arithmetic Sequence

an = a₁ (rⁿ⁻¹)

Formula to find the nth term in a geometric series

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 = n/2 (a1 +an)

Formula to find the sum of an arithmetic Sequence

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

d = an - a(n-1) .

Formula to find the common difference

r = an / a(n-1) .

Formula to find the common ratio