Problem 51 - Precalculus with Limits

Question

In Exercises 51 and 52, determine whether the statement is true or false. Justify your answer. The sum of the first positive integers is \( \frac{n(n+1)}{2} \).

Step-by-Step Solution

Answer

The statement 'The sum of the first positive integers is \( \frac{n(n+1)}{2} \)' is true.

Step 1: Understand the Statement

The statement is given as: The sum of the first n positive integers is \( \frac{n(n+1)}{2} \). The sum of first n positive integers would be represented as: 1 + 2 + 3 + .... + n. The statement suggests this sum equals \( \frac{n(n+1)}{2} \), which needs to be proven true or false.

Step 2: Mathematical Proof

Assume by mathematical induction that the statement is true for a certain positive integer k. Therefore: 1 + 2 + 3 + .... + k = \( \frac{k(k+1)}{2} \). Now for k + 1, it would be 1 + 2 + 3 + .... + k + (k + 1). Substituting the value from the induction hypothesis, this becomes \( \frac{k(k+1)}{2} \) + (k + 1). Simplifying this would give \( \frac{(k+1)(k+2)}{2} \), which is the correct form to confirm the statement is true.

Step 3: Conclusion

As verified using mathematical induction, when the formula \( \frac{n(n+1)}{2} \) is applied for any positive integer n, it correctly gives the sum of the first n positive integers. Therefore, the statement is indeed true.

Key Concepts

Mathematical Induction

Mathematical induction is a powerful method of proof used to establish the validity of statements for all positive integers. It consists of two main steps:

• Base Case: Confirm the statement is true for the first integer, usually n = 1.
• Inductive Step: Assume the statement is true for a particular integer k and then prove it is true for the next integer, k + 1.

In the case of proving the formula for the sum of the first n positive integers, we start by checking if the formula works for n = 1. For n = 1, the sum is 1, and the formula gives \( \frac{1\times(1+1)}{2} = 1 \), confirming the base case.

Next, assume the formula holds for n = k. This means 1 + 2 + ... + k = \( \frac{k(k+1)}{2} \). Then, for n = k + 1, the sum becomes 1 + 2 + ... + k + (k + 1). By substituting the formula, this sum becomes \( \frac{k(k+1)}{2} + (k + 1) \). Simplifying gives \( \frac{(k+1)(k+2)}{2} \), showing the formula holds for k + 1. Hence, by mathematical induction, the formula is true for all positive integers n.

Arithmetical Series

An arithmetical series is a sequence of numbers in which each term after the first is obtained by adding a constant, known as the common difference, to the previous term. For the first n positive integers, this series is: 1, 2, 3, ..., n.

The formula \( \frac{n(n+1)}{2} \) is derived from this concept. It efficiently calculates the sum of a linear sequence of increasing numbers, where each term increases by 1 from the previous term. This series is notable because

• it has a well-established formula for summation,
• the sum grows quadratically with n, since each new integer added increases the total by an incrementally larger margin.

This makes arithmetical series and their summation properties crucial in mathematical analysis and everyday computations.

Algebraic Proof

Algebraic proof involves using algebraic manipulation to show that a certain statement or formula is valid. This method often includes rearranging and simplifying equations to highlight certain truths.

For the sum of the first n positive integers, an algebraic proof can be used to reaffirm the induction conclusion. Consider the expression for the sum:

• The sequence 1 + 2 + ... + n can be mirrored to form a pair-wise addition, like \((1+n), (2+(n-1)), ..., (n+1-1)\). Each pair sums to n+1, and there are \( \frac{n}{2} \) such pairs if n is even (or approximately \( \frac{n}{2} \) pairs with a middle number, n/2, if n is odd).
• This visualization uses algebra to validate that the sum indeed matches the formula \( \frac{n(n+1)}{2} \).

Through logical sequencing and substitution, algebraic proofs provide a concrete understanding of why the mathematical statements hold true.