## Sequences and series

The program calculates the first n terms of a sequence  (ai)  and the corresponding 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 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. ```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)```