Sequences and Series

The program determines the first n terms of a sequence  (ai)  and the associated series (sum of the sequence terms) if the first terms of the sequence and an explicit function  ai=ƒ(i)  or 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 :

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

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

Serie
¯¯¯¯
( 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 proven very nicely by complete induction.

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.

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

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