Sequences and series

The program calculates the first n terms of a sequence  (ai)  and the corresponding series Σai (sum of the sequence terms) if an explicit function  ai=ƒ(i)  or the first terms of the sequence and a recourse formula  ai=ƒ(a0, a1, ... , ai-1)  are given.

The sequence of odd numbers

It can be defined explicitly by   ai = 2·i + 1 :

a[0] ... =

a[ i ] = 2*i + 1

n = 10

a[i] = 2*i + 1;   n = 10;	

Sequence
¯¯¯¯¯¯¯¯
( a[ i ] ) = (1; 3; 5; 7; 9; 11; 13; 15; 17; 19)

Series
¯¯¯¯¯
( Σ a[ i ] ) = (1; 4; 9; 16; 25; 36; 49; 64; 81; 100)

or recursively by  ai = ai-1 + 2  with  a0=1 .

a[0] ... = 1

a[ i ] = a[i-1]+2

n = 10

a[0] = 1;   a[i] = a[i-1]+2;   n = 10;	

Sequence
¯¯¯¯¯¯¯¯
( a[ i ] ) = (1; 3; 5; 7; 9; 11; 13; 15; 17; 19)

Series
¯¯¯¯¯
( Σ a[ i ] ) = (1; 4; 9; 16; 25; 36; 49; 64; 81; 100)

The corresponding series is obviously the sequence of the square numbers. This can be proved very nicely by complete induction. ( Wikipedia: Mathematical induction / Sum of consecutive natural numbers )

The Fibonacci Sequence

One of the most popular recursive sequences starts with  a0=1  and  a1=1 . The other terms are equal to the sum of the previous two.

a[0] ... = 1; 1

a[ i ] = a[i-1]+a[i-2]

n = 20

a[0]=1;   a[1]=1;    a[ i ] = a[i-1]+a[i-2];   n = 20

Sequence
¯¯¯¯¯¯¯¯
( a[ i ] ) = (1; 1; 2; 3; 5; 8; 13; 21; 34; 55; 89; 144; 233; 377; 610; 987; 1597; 2584; 4181; 6765)

Series
¯¯¯¯¯
( Σ a[ i ] ) = (1; 2; 4; 7; 12; 20; 33; 54; 88; 143; 232; 376; 609; 986; 1596; 2583; 4180; 6764; 10945; 17710)

See also:

Wikipedia: Sequence | Fibonacci number