Alternate proof to induction for integer sum  Precalculus  Khan Academy  Summary and Q&A
TL;DR
There is a simpler algebraic proof for the sum of positive integers, n times n plus 1 over 2.
Questions & Answers
Q: What was proven in the previous video using induction?
In the previous video, it was proven that the sum of all positive integers up to and including n can be expressed as n times n plus 1 over 2 using induction.
Q: What is the alternative proof presented in this video?
The alternative proof presented in this video uses pure algebra to show that the formula for the sum of positive integers is n times n plus 1 over 2, without relying on induction.
Q: How is the function S of n defined?
The function S of n is defined as the sum of all positive integers up to and including n, represented by 1 plus 2 plus 3 plus ... plus n.
Q: How is the algebraic proof constructed?
The proof involves rearranging the terms in the sum and adding the rows together. By adding the terms in a specific order, it is shown that the sum is equal to n times n plus 1 over 2.
Summary & Key Takeaways

In the previous video, the sum of all positive integers up to and including n was proven using induction.

This video presents a different proof for the same sum of positive integers without using induction.

By defining the function S of n as the sum of positive integers up to and including n, the proof simplifies algebraically.