Question

If the arithmetic mean of the first \(n\) natural numbers is 15, then \(n\) is _____ (1) 15 (2) 30 (3) 14 (4) 29

Short answer

Answer: The value of \(n\) is 29.

Step-by-step solution

Formula for the arithmetic mean

The arithmetic mean of a set of numbers is defined as the sum of the numbers divided by the total number of numbers. In this case, the set of numbers is the first \(n\) natural numbers, so we can write the arithmetic mean as: $$\text{Arithmetic Mean} = \frac{\text{Sum of first } n \text{ natural numbers}}{n}$$

Formula for the sum of the first \(n\) natural numbers

The sum of the first \(n\) natural numbers can be found using the formula: $$\text{Sum of first } n \text{ natural numbers} = \frac{n(n+1)}{2}$$

Set up the equation involving \(n\)

Given that the arithmetic mean of the first \(n\) natural numbers is 15, we can set up an equation by plugging in the formula for the sum of the first \(n\) natural numbers into the arithmetic mean formula: $$15= \frac{\frac{n(n+1)}{2}}{n}$$

Solve the equation for \(n\)

Now we can solve the equation to find the value of \(n\). First, multiply both sides by \(2n\) to simplify: $$15(2n) = n(n+1)$$ $$30n = n^2 + n$$ Next, move all the terms to one side and set the equation to zero: $$n^2 - 29n = 0$$

Factor and find the possible values for \(n\)

Factor the quadratic equation: $$n(n - 29) = 0$$ The possible values for \(n\) are: $$n = 0, n = 29$$ Since we are dealing with natural numbers, we can eliminate \(n=0\). Therefore, the correct answer is: $$n = 29$$ Thus, the correct option is (4) 29.

Key concepts

Arithmetic Mean Formula

The arithmetic mean, often referred to as the average, is a quick way to find a central value among a set of numbers. It is calculated by adding up all the numbers in a set and then dividing by the count of numbers. In mathematical terms, if you have a set of numbers, let's call them \( x_1, x_2, ..., x_n \), the arithmetic mean (AM) is given by the formula:
\[ AM = \frac{\sum_{i=1}^n x_i}{n} \]
This formula is particularly useful when dealing with a series of consecutive natural numbers. Natural numbers are those that begin from 1 and go up to infinity, and they are often the first type of numbers that students learn about. Understanding how to compute the arithmetic mean of such a series is a vital skill in mathematics and statistics.

Sum of Natural Numbers

The sum of natural numbers can be thought of as the result of adding all the numbers from 1 to a certain number \( n \). There's an elegant formula to calculate this sum without having to add each number individually, which was famously discovered by the mathematician Carl Friedrich Gauss as a child. The formula is:
\[ S_n = \frac{n(n + 1)}{2} \]
This formula indicates that if you want to find the sum of the first \( n \), you multiply \( n \), the last number in the set, by \( n + 1 \), the next number, and then divide by 2. It is an application of a simple arithmetic sequence where each number is one more than the previous number. For example, to calculate the sum of the first 10 natural numbers, we use the formula to get \( S_{10} = \frac{10(10 + 1)}{2} = 55 \). Understanding how to apply this formula is essential for a range of mathematical problems, including those in algebra and calculus.

Solving Quadratic Equations

Quadratic equations are a step up from linear equations and have the general form \( ax^2 + bx + c = 0 \) where \( a \), \( b \), and \( c \) are constants. Solving a quadratic equation means finding the values of \( x \) that make the equation true. There are several methods to solve these, including factoring, completing the square, and using the quadratic formula.

Factoring

This involves rewriting the quadratic as a product of two binomials. If the quadratic can be factored, it is written as \( (x - p)(x - q) = 0 \) where \( p \) and \( q \) are the solutions to the equation.

Quadratic Formula

When factoring is difficult or not possible, the quadratic formula: \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) can be used, providing a guaranteed way to find the roots of any quadratic equation. In most academic settings, it's imperative that students learn to solve quadratic equations, as they are fundamental in various areas of mathematics and science.