Problem 60 - Algebra and Trigonometry

Question

Solve the equation for the indicated variable. $$ S=\frac{n(n+1)}{2} ; \text { for } n $$

Step-by-Step Solution

Answer

The solution for \( n \) is \( n = \frac{-1 + \sqrt{1 + 8S}}{2} \).

Step 1: Understand the Equation

The given equation is a formula for the sum of the first \( n \) natural numbers, represented by \( S = \frac{n(n+1)}{2} \). We need to solve this equation for \( n \).

Step 2: Clear the Fraction

To eliminate the fraction, multiply both sides of the equation by 2:\[2S = n(n + 1)\]

Step 3: Expand and Rearrange

The right side of the equation is a product. Expand it:\[2S = n^2 + n\]Rearrange this equation into a standard quadratic form:\[n^2 + n - 2S = 0\]

Step 4: Apply the Quadratic Formula

Solve the quadratic equation \( n^2 + n - 2S = 0 \) using the quadratic formula \( n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 1 \), \( b = 1 \), \( c = -2S \):\[n = \frac{-1 \pm \sqrt{1^2 - 4 \cdot 1 \cdot (-2S)}}{2 \cdot 1}\]Simplify it:\[n = \frac{-1 \pm \sqrt{1 + 8S}}{2}\]

Step 5: Select the Positive Solution

Since \( n \) must be a positive integer (as it represents a count), choose the positive root:\[n = \frac{-1 + \sqrt{1 + 8S}}{2}\]

Key Concepts

Quadratic Formula

The quadratic formula is a powerful tool for solving quadratic equations. These are equations of the form \( ax^2 + bx + c = 0 \). It provides a precise method to find the values of \( x \) that satisfy this equation. The quadratic formula is given by:

• \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)

The components of this formula include:

• \( a \), \( b \), and \( c \): coefficients from the quadratic equation.
• \( \Delta = b^2 - 4ac \): often called the discriminant, reveals information about the nature of roots.

The formula helps determine:

• Real and distinct roots when \( \Delta > 0 \)
• Real and repeated roots when \( \Delta = 0 \)
• Complex roots when \( \Delta < 0 \)

This formula is especially useful when factoring is difficult or impossible, making it a staple in solving quadratic equations.

Natural Numbers

Natural numbers are the set of positive integers starting from 1 upwards. They are often represented by the symbol \( \mathbb{N} \) and play a foundational role in arithmetic and number theory. Some distinguishing characteristics include:

• Starting at 1 and increasing by increments of 1.
• Representing quantities like counting numbers and ordered positions.

For example, the series \( 1, 2, 3, 4, \ldots \) is infinitely ongoing. Natural numbers do not include zero or any negative numbers. Additionally, they are used to define more complex concepts such as sums, factors, and sequences. Their simplicity allows them to be one of the first number systems learned.

Sum of Natural Numbers

The sum of the first \( n \) natural numbers is defined by the formula \( S = \frac{n(n+1)}{2} \). This equation gives a quick way to add up a series of consecutive natural numbers starting from 1.

• \( S \) represents the total sum.
• \( n \) is the number of terms you are summing up.

The derivation of this formula can be visualized by pairing numbers symmetrically across the sequence. For instance, in the sum \( 1 + 2 + 3 + \ldots + n \), we can rearrange terms as \( (1+n) + (2+(n-1)) + \ldots \), each pair summing to the same value and simplifying the calculation process.
This principle not only showcases the efficiency of mathematical shortcuts but also illustrates the elegance and symmetry inherent in arithmetic structures. It's widely used in various mathematical problems and applications.

Positive Integers

Positive integers are all the whole numbers greater than zero. They are represented as \( 1, 2, 3, \ldots \) and are intrinsic in arithmetic operations and number theory. Key features of positive integers include:

• Used for counting ("how many?") purposes.
• Lack of decimals or fractions; they are whole numbers.

Within expressions and equations, positive integers often specify counts, such as quantities or positions within a set or a sequence. They form an infinite set in mathematics, acting as the backbone of numerical operations, and they interconnect with various other number systems like rational numbers and real numbers to build complex mathematical frameworks.
In context, when solving an equation for \( n \), as in our task, \( n \) is naturally a positive integer because it represents a count, such as the number of terms.