Problem 25
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
Verified Answer
\( S = \frac{n(n+1)}{2} \) shows the sum of the series as derived by adding two forms of the series.
1Step 1: Identify the Problem
We are given two equal expressions for \( S \). The goal is to add these two equations and simplify the expression to find an alternative proof for the sum of the series.
2Step 2: Write Down Both Equations
The first equation is \( S = 1 + 2 + 3 + \cdots + (n-1) + n \). The second equation, written in reverse order, is \( S = n + (n-1) + (n-2) + \cdots + 2 + 1 \).
3Step 3: Add the Equations
Add the two equations term by term. The sum of each pair from the two equations is \( (1+n), (2+(n-1)), (3+(n-2)), \ldots, (n+1) \). Hence, adding the equations together results in \( 2S = (1+n) + (2+(n-1)) + (3+(n-2)) + \cdots + (n+1) \).
4Step 4: Recognize the Pattern
Observe that each pair of terms, when added, equals \( n+1 \). Thus, there are \( n \) pairs, and each sums to \( n+1 \).
5Step 5: Simplify the Summation
Calculate the total for all pairs: \( 2S = n(n+1) \) because there are \( n \) pairs each summing to \( n+1 \).
6Step 6: Solve for S
Divide both sides by 2 to isolate \( S \): \[ S = \frac{n(n+1)}{2} \]. This provides another proof of the formula for the sum of the first \( n \) natural numbers.
Key Concepts
Sum of seriesNatural numbersMathematical proof
Sum of series
The sum of a series refers to the result of adding a sequence of numbers. In arithmetic series, the numbers in the sequence increase by a constant difference. For instance, the sequence 1, 2, 3,..., n is an arithmetic series where each number increases by 1. Understanding how to find the sum of such a series is fundamental in mathematics.
To find the sum of the first n natural numbers, we use the formula \[S = \frac{n(n+1)}{2}\] where \( S \) is the sum, and \( n \) represents the last number in the sequence.
In the provided exercise, the two sequences are summed in different orders, yet they ultimately represent the same total. By adding these sequences term by term, you recognize a pattern: each term pair sums to \( n+1 \). Ultimately, there are \( n \) terms that each add up to this value, leading to a total of \( n(n+1) \) before dividing by 2 to solve for \( S \). This demonstrates the elegance and consistency of arithmetic sequences.
To find the sum of the first n natural numbers, we use the formula \[S = \frac{n(n+1)}{2}\] where \( S \) is the sum, and \( n \) represents the last number in the sequence.
In the provided exercise, the two sequences are summed in different orders, yet they ultimately represent the same total. By adding these sequences term by term, you recognize a pattern: each term pair sums to \( n+1 \). Ultimately, there are \( n \) terms that each add up to this value, leading to a total of \( n(n+1) \) before dividing by 2 to solve for \( S \). This demonstrates the elegance and consistency of arithmetic sequences.
Natural numbers
Natural numbers are the basic counting numbers starting from 1 and progressing infinitely: 1, 2, 3, 4, ... These numbers exclude zero and negative numbers, making them useful for counting and ordering.
In the context of sum of series, particularly arithmetic ones, natural numbers often form the simplest and most illustrative examples. The series 1, 2, 3,..., n shows how natural numbers increase by a fixed amount, known as the common difference, which in this case is 1.
Working with natural numbers to understand series sums also lays a foundation for more complex mathematical concepts. By recognizing patterns and formulating simple algebraic expressions, learners can build a strong base for advanced mathematics.
In the context of sum of series, particularly arithmetic ones, natural numbers often form the simplest and most illustrative examples. The series 1, 2, 3,..., n shows how natural numbers increase by a fixed amount, known as the common difference, which in this case is 1.
Working with natural numbers to understand series sums also lays a foundation for more complex mathematical concepts. By recognizing patterns and formulating simple algebraic expressions, learners can build a strong base for advanced mathematics.
Mathematical proof
Mathematical proof is the process of demonstrating that a certain statement or equation is universally true using logical reasoning. Proofs involve a series of logical steps and are critical in establishing integrity and consistency within mathematics.
In the exercise, we see a proof by adding two expressions of an arithmetic series in different orders. This technique is not just about calculating but also about understanding relationships between numbers.
In the exercise, we see a proof by adding two expressions of an arithmetic series in different orders. This technique is not just about calculating but also about understanding relationships between numbers.
- Write down the sequence in a standard order and a reverse order.
- Each pair of terms across the sequences add up to the same constant value.
- Calculate the total sum using recognized patterns.
Other exercises in this chapter
Problem 25
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