3.4 The Sum of Sequences
Imagine that you are going to add the numbers 1, 2, 3, 4, 5 and so forth all the way up to 98, 99 and 100.
One way is to add the numbers with each other the entire way up to one hundred. It will take a little time though. This exercise was given to Carl Friedrich Gauss (17771855) and his classmates by their math teacher in school.
It was then that Carl Friedrich solved the problem by first setting up the numbers like this:
If you add two of the numbers together, one from every row which is in the same column, then the sum becomes 101 as you can see here. How many 101’s will there be?

1 
+ 
2 
+ 
3 
+ 
... + 
98 
+ 
99 
+ 
100 

+ 
100 
+ 
99 
+ 
98 
+ 
... + 
3 
+ 
2 
+ 
1 




101 
+ 
101 
+ 
101 
+ 
... + 
101 
+ 
101 
+ 
101 

There will be one hundred 101’s. Because we have added every number in the sequence twice, we divide by two and in this way we get the solution:
The sum of the numbers 1 to 100 gives:
101 · (100 ÷ 2) = 101 · 50 = 5,050
The sum of the numbers
1 + 2 + 3 + ... + 98 + 99 + 100
= 101 · 50 = 5,050
The sum of the numbers 1 + 2 + 3... + 97 + 98 + 99 + 100 also be calculated with the formula:
f(n) = 
n · (n + 1) 
= 
n^{2} + n 


2 
2 
Where n is the number of terms in the sequence. 