Problem 93 - College Algebra with Modeling and Visualization

Question

Verify the formula \(\sum_{k=1}^{n} k=\frac{n(n+1)}{2}\) by using the formula for the sum of the first \(n\) terms of a finite arithmetic sequence.

Step-by-Step Solution

Answer

The formula is verified by using the sum of the first \( n \) terms of an arithmetic sequence.

Step 1: Identify the Arithmetic Sequence

The sequence of natural numbers from 1 to \( n \), i.e., \( 1, 2, 3, \, \ldots, n \), is an arithmetic sequence where each term increases by 1. It has the first term \( a = 1 \) and the last term \( l = n \).

Step 2: Apply the Formula for the Sum of an Arithmetic Sequence

The formula for the sum \( S_n \) of the first \( n \) terms of an arithmetic sequence is given by \( S_n = \frac{n}{2} (a + l) \), where \( a \) is the first term and \( l \) is the last term.

Step 3: Substitute Known Values

Substitute \( a = 1 \), \( l = n \), and the number of terms \( n \) into the formula: \[ S_n = \frac{n}{2} \times (1 + n) = \frac{n (1 + n)}{2} = \frac{n(n+1)}{2} \]

Step 4: Compare with the Given Formula

The derived expression \( \frac{n(n+1)}{2} \) matches the given formula \( \sum_{k=1}^{n} k = \frac{n(n+1)}{2} \), thus verifying its correctness.

Key Concepts

Sum of Finite Arithmetic Sequence

An arithmetic sequence, or arithmetic progression, is a sequence of numbers in which each term after the first is obtained by adding a constant, known as the common difference.
When dealing with an arithmetic sequence that eventually stops after a certain number of terms, it's termed as a finite arithmetic sequence.
To find the sum of a finite arithmetic sequence, we use the formula:

• \( S_n = \frac{n}{2} (a + l) \)

Here, \( S_n \) is the sum of the first \( n \) terms, \( a \) is the first term, \( l \) is the last term, and \( n \) is the number of terms.
This formula is convenient because it allows for quick calculation by simply averaging the first and last terms and multiplying by the number of terms.
This formula is particularly useful when faced with long sequences as it simplifies what could be a cumbersome process of adding each term individually.

Formula Verification

Verifying mathematical formulas is important to ensure that they work as intended under all circumstances.
In this context, we are tasked with proving the validity of the formula for the sum of the first \( n \) natural numbers:

• \( \sum_{k=1}^{n} k = \frac{n(n+1)}{2} \)

One way to verify this formula is to recognize that the sequence \( 1, 2, 3, \ldots, n \) is a finite arithmetic sequence.
We have

• \( a = 1 \) (the first term),
• \( l = n \) (the last term),
• and \( n \) is the total number of terms in the sequence.

By substituting these values into our formula for the sum of a finite arithmetic sequence, we derive \( \frac{n}{2} \times (1 + n) = \frac{n(n+1)}{2} \).
When our derived result matches the given formula, it confirms the formula's accuracy and reliability.

Series and Sequences

Sequences and series are fundamental concepts in mathematics.
A sequence is a list of numbers that follow a specific pattern, while a series is essentially the sum of the terms of a sequence.
Arithmetic sequences are just one type, defined by their constant difference between consecutive terms.
Mathematicians encounter sequences and series in various contexts, from analyzing patterns to solving complex equations.

• Arithmetic sequences are simple yet powerful, forming the foundation for more advanced concepts.
• They provide an understandable entry point into the realm of sequences because of their straightforward nature.

Understanding sequences and series allows for a deeper comprehension of algebraic structures and mathematical analysis.
Whether working out finance models or solving real-life problems, these concepts play a crucial role in simplifying understanding big numbers through patterns and summation.