Problem 25 - Calculus Early Transcendentals: Pearson New International Edition

Question

Add both sides of the two equalities below, solve for \(S\). and thereby give another proof of Formula \(1 .\) $$ \begin{array}{l} S=1+2+3+\cdots+(n-2)+(n-1)+n \\ S=n+(n-1)+(n-2)+\cdots+3+2+1 \end{array} $$

Step-by-Step Solution

Answer

The formula is \( S = \frac{n(n+1)}{2} \).

Step 1: Understanding the Given Equations

We are given two equal sums expressed in different forms. The first equation is \( S = 1 + 2 + 3 + \cdots + (n-1) + n \). The second equation is \( S = n + (n-1) + (n-2) + \cdots + 2 + 1 \). Both expressions represent the same sum, just written in reverse order.

Step 2: Add the Corresponding Terms of the Equations

Add the terms of both equations together: the first term of the first equation with the first term of the second equation, the second term with the corresponding term, and so on. This gives: \[ (1+n) + (2+(n−1)) + (3+(n−2)) + \cdots + ((n−1)+2) + (n+1) = S + S \] This simplifies to: \[ 2S = (n+1) + (n+1) + (n+1) + \cdots + (n+1) \], where there are \( n \) pairs, each summing to \( n+1 \).

Step 3: Simplifying the Summation

The equation \( 2S = n(n+1) \) results from the summation since there are \( n \) terms each equal to \( n+1 \). Thus, the right side becomes \( n \times (n+1) \).

Step 4: Solve for \( S \)

Divide both sides of the equation \( 2S = n(n+1) \) by 2 to solve for \( S \): \[ S = \frac{n(n+1)}{2} \]. This provides an alternative proof for the formula for the sum of the first \( n \) natural numbers.

Key Concepts

Arithmetic Series

An arithmetic series is a sequence of numbers in which the difference between consecutive terms is constant. In the case of adding natural numbers, the sequence starts at 1 and increases by 1 each time, forming a simple arithmetic series.
For example, the series 1, 2, 3,..., n can be seen as an arithmetic series where the first term is 1 and the common difference is also 1.
The sum of an arithmetic series can be calculated quickly using a formula, which simplifies the process and saves time, especially for large sequences.

Mathematical Proof

A mathematical proof is a logical argument that verifies the truth of a given statement or formula. In our scenario, establishing the formula for the sum of natural numbers can be affirmed using different proofs.
One common proof approach involves writing the sequence forwards and backwards, then adding them to find a pattern or relationship
Our proof utilized this clever method of rewriting the sum in reverse and adding paired terms, highlighting how mathematics can reveal consistent patterns.

Summation Formula

The summation formula for the sum of natural numbers is: \[ S = \frac{n(n+1)}{2} \] This formula provides a quick and efficient way to calculate the total sum of the first n natural numbers without needing to add each number individually.
Understanding and using this formula is important in mathematics for simplifying the process of finding the sum of any integer sequence that fits this arithmetic series.
It allows us to transition from a series of individual additions to a more structured algebraic calculation.

Integer Sequences

Integer sequences are lists of numbers where the elements are whole numbers, such as natural numbers, whole numbers, or even numbers.
In mathematics, sequences give us a way to arrange numbers in particular patterns or orders, making it easier to find relationships and rules.
The sum of natural numbers is a specific integer sequence that forms a simple and foundational pattern—each subsequent number adds 1 to the previous.
Identifying this order allows us to use mathematical formulas, like the summation formula, to describe sequences mathematically and find sums or other properties.