Sum of Integers from 1 to n

What does the sum of integers from 1 to $n$ mean?

Natural numbers are positive numbers starting from $1$. These can be written as:

$1,2,3,4,5,6,7,8,9,10......$

We want to write a program that takes a number and sums up all the numbers from $1$ up to that number.

For example, if $n = 5$, then the sum of numbers from $1$ to $5$ is: $1 + 2 + 3 + 4 + 5 = 15$.

Mathematical Notation

The sum of all numbers up to a number is equal to the sum of that number and the sum of all the numbers before it. Let’s write it down mathematically:

$\sum_{i=1} ^{5} i$

$=$ $5$ $+$ $\sum_{i=1} ^{4} i$

$=$ $5$ $+$ $4$ $+$ $\sum_{i=1} ^{3}$

$=$ $5$ $+$ $4$ $+$ $3$ $+$ $\sum_{i=1} ^{2}$

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.

.

$=5+4+3+2+1$

Generic Mathematical Notation

$\sum_{i=1} ^{n} i$

$=$ $n$ $+$ $\sum_{i=1} ^{n-1} i$

$=$ $n$ $+$ $(n-1)$ $+$ $\sum_{i=1} ^{n-2}$

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.

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$=$ $n$ $+$ $(n-1)$ $+(n-2)$ $ … +2+1$

Implementation