SEQUENCES & SERIES

Miguel Pérez Fontenla, Nov 2009

1,2,3,4,5,.....

1,2,3,4,5,.....

...6,7,8,......

{ }n

1,2,3,4,5,.....

...6,7,8,......

{ }n

1,2,3,4,5,.....

...6,7,8,......

{ }nNatural numbers

2,4,6,8,.....

2,4,6,8,.....

...10,12,14......

2,4,6,8,.....

...10,12,14......

{ }2n

2,4,6,8,.....

...10,12,14......

Even numbers

{ }2n

1,3,5,7,.....

1,3,5,7,.....

...9,11,13......

1,3,5,7,.....

...9,11,13......

{ }2 1n −

1,3,5,7,.....

...9,11,13......

Odd numbers

{ }2 1n −

4,7,10,13,.....

4,7,10,13,.....

...16,19,22......

4,7,10,13,.....

...16,19,22......

{ }3 1n +

4,7,10,13,.....

...16,19,22......

Multiple of three plus one

{ }3 1n +

2,4,8,16,.....

2,4,8,16,.....

...32,64,128.....

2,4,8,16,.....

...32,64,128.....

{ }2n

2,4,8,16,.....

...32,64,128.....

Powers of two

{ }2n

1,2,4,8,16,.....

1,2,4,8,16,.....

...32,64,128.....

1,2,4,8,16,.....

...32,64,128.....

{ }12n−

1,2,4,8,16,.....

...32,64,128.....

Powers of two

{ }12n−

1,3,9,27,.....

1,3,9,27,.....

...81,243,.....

1,3,9,27,.....

...81,243,.....

{ }13n−

1,3,9,27,.....

...81,243,.....

Powers of three

{ }13n−

2,5,10,17,.....

2,5,10,17,.....

...26,37,.....

2,5,10,17,.....

...26,37,.....

{ }2 1n +

2,5,10,17,.....

...26,37,.....

n squared plus one

{ }2 1n +

1,8,27,64,.....

1,8,27,64,.....

...125,216,.....

1,8,27,64,.....

...125,216,.....

{ }3n

1,8,27,64,.....

...125,216,.....

n cubed

{ }3n

2,6,12,20,.....

2,6,12,20,.....

...,30,42,56,.....

2,6,12,20,.....

...,30,42,56,.....

( ){ }1n n +

2,6,12,20,.....

...,30,42,56,.....

n multiplied by n plus one

( ){ }1n n +

1 1 3, , ,....

2 2 2

−

1 1 3, , ,....

2 2 2

−

5 7 9..., , , ,....

2 2 2

1 1 3, , ,....

2 2 2

−

2 3

2

n −

5 7 9..., , , ,....

2 2 2

1 1 3, , ,....

2 2 2

−

2 3

2

n −

5 7 9..., , , ,....

2 2 2

two times n minus three divided by two

-1,1,-1,1,.....

...-1,1,-1,1,.....

minus one to the power of n

( ){ }1n−

1,-1,1,-1,.....

1,-1,1,-1,.....

...1,-1,1,-1,.....

1,-1,1,-1,.....

...1,-1,1,-1,.....

( ){ }11n+−

1,-1,1,-1,.....

...1,-1,1,-1,.....

minus one to the power of n plus one

( ){ }11n+−

2 4 6 8, , , ....

1 3 5 7

2 4 6 8, , , ....

1 3 5 7

10 12 14..., , , ,....

9 11 13

2 4 6 8, , , ....

1 3 5 7

10 12 14..., , , ,....

9 11 13

2

2 1

n

n −

two times n divided by two times n minus one

2 4 6 8, , , ....

1 3 5 7

10 12 14..., , , ,....

9 11 13

2

2 1

n

n −

and....Other series not so easy

1,0,1,0,1.....

1,0,1,0,1.....

...0,1,0,1.....

1,0,1,0,1.....

...0,1,0,1.....

( ) 11 if n odd 1 1

0 if n even 2

n

na++ −

= =

1,1,2,3,5.....

1,1,2,3,5.....

...8,13,21,.....

Fibonacci Sequence

1,1,2,3,5.....

...8,13,21,.....

Fibonacci Sequence

1 2

1 if n = 1

if n > 1nn n

aa a− −

= +

81,27,9,.....

81,27,9,.....

...3,1,1/3,.....

81,27,9,.....

...3,1,1/3,.....

5

1

3n−

81,27,9,.....

...3,1,1/3,.....

on divided by 3 to the power of n minus five

5

1

3n−

2,3,5,7,11,.....

2,3,5,7,11,.....

...13,17,19,23.....

prime numbers

0,1,4,6,8,9,10,.....

0,1,4,6,8,9,10,.....

...12,14,15,16.....

no prime numbers

2,-4,6,-8.....

2,-4,6,-8.....

...10,-12,14,...

2,-4,6,-8.....

...10,-12,14,...

( ){ }11 2n

n+−

2,-4,6,-8.....

...10,-12,14,...

minus one to the power of n plus one multiplied by two times n

( ){ }11 2n

n+−

81 27 9 3, , , ,...

16 8 4 2

two thirds to the power of n minus five

81 27 9 3, , , ,...

16 8 4 2

52 4 8 2

1, , , ...3 9 16 3

n−

1 2, , 2,...

4 2

1 2, , 2,...

4 2

( ) 1

2 24 2,16,32 2,...

4

n−

two tine 2 squared to the power of n minus one divided by four

1 2, , 2,...

4 2

( ) 1

2 24 2,16,32 2,...

4

n−

and now....

Funny series

a,b,d,g,k.....

a,b,d,g,k.....

...p,v,...

following the sequence 1,2,4,7,11,16,22

O,T,T,F,F,S,S.....

O,T,T,F,F,S,S.....

...,E,N,T

O,T,T,F,F,S,S.....

...,E,N,T

One, two, three, four, five, six, seven, eight, nine, ten....

M,T,W,T,F.....

M,T,W,T,F.....

...S,S

Week days

M,V,E,M,J.....

M,V,E,M,J.....

...S,N,U,P

Planets Solar SystemMercury, Venus, Earth, Mars, Jupiter, Saturn, Neptune, Uranus,

Pluto

F,S,T,F,F,S,S.....

F,S,T,F,F,S,S.....,E,N,T

J,F,M,A,M,J,J.....

J,F,M,A,M,J,J........,A,S,O,N,D

31,28,31,30,31,...

31,28,31,30,31,...

...30,31,31,..Lengths of months in days

2,2,4,4,2,6,6,2,8,...

2,2,4,4,2,6,6,2,8,...

...8,2,10,...Table of two

5,4,2,9,8,6,...

5,4,2,9,8,6,...

...7,3,1Alphabetic Order In Spanish

2,10,12,16,17,18,19,200...

2,10,12,16,17,18,19,200...

...201,202,..

In Spanish they start by the “D” letter:Dos, diez, doce, dieciseis,

diecisiete,dieciocho,diecinueve,doscientos....

2,3,6,16,22,23,26,32,33...

2,3,6,16,22,23,26,32,33...

...36,42,43,46,..

In Spanish they finish with the “S” letter:Dos, tres, seis, dieciseis, veintidos, veintitres, veintiseis,

treinta y dos ....