Sequences And Series

Arithmetic

Arithmetic Sequence

An arithmetic sequence
is a sequence where a
common difference is
added to each consecutive
term.

Examples :
1, 2, 3, 4, 5 d = 1
16, 33, 50, 67, 84 d = 17
1, -3, -7, -11, -15 d = -4

Formula :
An = A1 + (n-1) d

Arithmetic Series

An arithmetic series
is the sum of all the
terms of an arithmetic
sequence.

Examples :
1, 2, 3, 4, 5 S5 = 15
16, 33, 50, 67, 84 S5 = 250
1, -3, -7, -11, -15 S5 = -35

Formula :
Sn = n/2 (A1+An)

Geometric Sequence

A geometric sequence is
a sequence where a common
ratio is multiplied to each
consecutive term.

Examples :
1, 2, 4, 8, 16 r = 2
1, 3, 9, 27, 81 r = 3
1, 5, 25, 125, 625 r = 5

Formula :
An = A1r^n-1

Geometric Series

A geometric series is
the sum of all the terms
of a geometric sequence.

Examples :
1, 2, 4, 8, 16 S5 = 31
1, 3, 9, 27, 81 S5 = 121
1, 5, 25, 125, 625 S5 = 781

Formula : (Finite)
Sn = A1 (1-r^n)
(-------)
( 1-r )
or
Sn = A1 (r^n-1)
(-------)
( r-1 )
Formula : (Infinite)
Sn = A1/r-1

A harmonic sequence is
a sequence where the
reciprocals of the terms
form an arithmetic sequence.

Examples :
1, 1/2, 1/3, 1/4, 1/5
1, 1/16, 1/33, 1/50, 1/67, 1/84
1, -1/3, -1/7, -1/11, -1/15

A fibonacci sequence is
a sequence where the
previous term is added
to the next term.

Example :
0, 1, 1, 2, 3,
5, 8, 13, 21, 34,
55, 89, 144, 233, 377