The Formula for 1+2+3+4+5 to 10000: A Mathematical Exploration
Have you ever wondered about the formula for calculating the sum of consecutive integers from 1 to a large number, such as 10000? In this article, we will delve into the fascinating world of mathematics and explore the formula behind this intriguing sequence.
The Problem Statement
Given a series of consecutive integers from 1 to n, where n is a large number, what is the formula to calculate the sum of these integers?
The Formula
The formula to calculate the sum of consecutive integers from 1 to n is given by:
1 + 2 + 3 + ... + n = n(n+1)/2
This formula is known as the formula for the sum of an arithmetic series.
Derivation of the Formula
To derive this formula, let's consider the sequence of consecutive integers from 1 to n:
1 + 2 + 3 + ... + n
We can write this sequence as:
(1) + (2) + (3) + ... + (n)
Now, let's reverse the order of the sequence:
n + (n-1) + (n-2) + ... + 1
When we add the two sequences, we get:
(n+1) + (n+1) + (n+1) + ... + (n+1)
Since there are n terms in the sequence, we can write:
n(n+1)
Now, dividing both sides by 2, we get:
n(n+1)/2
which is the formula for the sum of consecutive integers from 1 to n.
Example
Let's calculate the sum of consecutive integers from 1 to 10000 using the formula:
n = 10000
Plugging in the value of n, we get:
1 + 2 + 3 + ... + 10000 = 10000(10001)/2 = 50,000,500
Conclusion
In this article, we have explored the formula for calculating the sum of consecutive integers from 1 to n. We have derived the formula using a simple and intuitive approach, and demonstrated its application with an example. The formula n(n+1)/2 is a powerful tool for calculating the sum of large sequences of consecutive integers.
