Proof of the sum of the series ∑r

Proof of the sum of the series ∑r

Sum of the Series ∑r

Commencing the Proof

• The series we are trying to sum is the arithmetic series given by 1 + 2 + 3 + … + n. This is represented in Sigma notation as ∑r from r=1 to n.
• To sum this series, we begin with the old math trick of writing the series forwards and backwards and then adding them together.

Arranging the Series

• Write down the series as S = 1 + 2 + 3 + … + (n-2) + (n-1) + n.
• Then write it again backwards as S = n + (n-1) + (n-2) + … + 3 + 2 + 1.

Formulating an Equation

• Add these two expressions together: you get 2S = (n + 1) + (n + 1) + … + (n + 1), n times over.
• This means 2S = n(n + 1) because we are adding together (n + 1), n times.

Solving for S (Sum of the Series ∑r)

• From the expression 2S = n(n + 1), you can see that the sum S of the series 1 + 2 + 3 + … + n is given by S = n(n + 1) / 2.

Verification of the Proof

• This equation can be verified by substituting values for n and checking that both sides of the equation match.
• For example, if n = 5, the left side yields 1 + 2 + 3 + 4 + 5 = 15, and the right side yields 5(5 + 1) / 2 = 15. Since both sides match, the result is valid.

Understanding the Basic Concept

• This proof is fundamental as it forms the foundation for understanding more complex series and sequences in further studies.
• It is a clever piece of mathematical reasoning that combines the properties of numbers arranged in specific sequences (series) with algebraic techniques ** to obtain a concise formula for summing all the numbers from **1 to n.