Today we look at a few summation formulas and properties.
Given a sequence a1, a2, ... of numbers, the finite sum a1 + a2 + ... +an where n is a positive number
This sum can be written as n(n+1)/2 when the terms are consecutive n numbers and its terms can be added in any order. This is called the arithmetic series and has an order of n^2. If the series is infinite and a limit exists then it is said to converge otherwise it is said to diverge. We can rearrange the terms of an absolutely convergent series.
For any real number c and any finite sequences a1, a2 ... am and b1, b2 ... bn
Sum  c.ak + bk where k can range from 1 to n is equivalent to writing
c.Sum ak + Sum bk and this is said to be the linearity property.
The linearity property is also obeyed by infinite convergent series.
Summation of squares and cubes also have well defined sums.
For example Sum of squares of consecutive 1 to n numbers is n(n+1)(2n+1)/6
and the sum of cubes of consecutive 1 to n numbers is n^2(n+1)^2 / 4
A series is said to be geometric or exponential when the terms are increasing in power such as 1 + x + x^2 + x^3 ... +x^n and x is not equal to  1.
The sum of this series has the value (x^(n+1)  - 1)/(x-1)
When the summation is infinite and the absolute value of x is less than 1, then we have the sum as 1 / (1-x)
A Harmonic series is one where the nth harmonic number is the sum of the consecutive fractions 1 + 1/2 + 1/3 + ... + 1/n which we calculate as ln n + a constant
Additional formulas can be obtained by integrating and differentiating the series above.
Telescopic series is one where for any sequence a0, a1, ..., an,
Sum for k = 1 to n (ak - ak-1) = an - a0
since each of the terms a1, a2, ..., an-1 is added in  exactly once and subtracted out exactly once. Hence the term telescopes.

#codingexericse 

Given a representation of all distinct digits in an alien system, Generate an increasing sequence from 1 to 9 

For example  given 01 and 0F8 as the alien number system 

0 = 0           0           F, 8, F0, FF, F8, 80, 8F, 88, F00, F0F 

1  = 1          F 

2  = 10        8 

3  = 11        F0 

4 = 100        FF 

5 = 101       F8 

6 =  110      80 

7  = 111      8F 

8 =  1000      88 

9 =  1001      F00 

10 = 1010    F0F 

                     F08 

                     FF0 

                     FFF 

                     FF8 

                     F80 

                     F8F 

                     F88 

                     888 

                   F000 

void GenSequence(String pattern, int number) 

{ 

char[] array = pattern.ToCharArray();  

int len = array.Length; 

String prefix = string.empty; 

while (number) 

{ 

   int rem = number%len; 

   prefix += array[rem]; 

   number = number / len; 

} 

Console.WriteLine(prefix.reverse()); 

}