Math ABC - Discrete Functions 03

What is Series?

Sum of the terms of a sequence.

What is Arithmetic Series?

The sum of the first n terms of an arithmeic sequence is Arithmetic Series

  Sn = a + (a + d) + (a + 2d) + (a + 3d) + ... + (tn -d) + tn

After the following calcutions, we can get other fomulas for Sn:

  _(Note: tn is the n(th) term in the arithmetic sequence)

 ①  Sn =  a + (a + d) + (a + 2d) + (a + 3d) + ... + (tn - d) + tn
 
 ②  Sn =  tn + (tn - d) + (tn - 2d) + ... + (a + 3d) + (a + 2d) + (a + d) + a

 ① + ② = 2Sn = n * (tn + a)

So:

  Sn = n * (tn + a) / 2 

Also Because:

  tn = a + (n - 1)d

So:

 Sn = n * (a + (n - 1)d + a) / 2 = n * (2a + (n - 1)d) / 2

What is Geometric Series?

The sum of the first n terms of an arithmeic sequence is Geometric Series.

  Sn = a + ar + ar² + ar³ + ... ar^n

Let's do some calculations;

  ①  Sn = a + ar + ar² + ar³ + ... ar^(n-1)

  ②  r * Sn = ar + ar² + ar³ + ... + ar^(n-1) + ar^n

  ② - ① 

  (r-1)Sn = ar^n - a
 

  So: S(n) = (ar^n - a)/(r - 1)   (r <> 1)

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There are total 2 posts about Discrete Functions:

Math ABC - Discrete Functions 01

Math ABC - Discrete Functions 02

Math ABC - Discrete Functions 03