Question

Find the arithmetic mean of the first 567 natural numbers. (1) 284 (2) \(283.5\) (3) 283 (4) None of these

Short answer

(a) 284 (b) 283.5 (c) 285 (d) None of these Answer: (d) None of these (the arithmetic mean is calculated to be 284.5)

Step-by-step solution

Write down the formula for the sum of the first n natural numbers

We will use the formula Sum = n(n + 1) / 2, where n is the number of terms. In our case, n = 567.

Calculate the sum

Now, we insert the value of n (567) into the sum formula: Sum = 567(567 + 1) / 2 Sum = 567(568) / 2 Sum = 161356

Find the arithmetic mean

To find the arithmetic mean, we divide the sum by the number of terms: Arithmetic Mean = Sum / n Arithmetic Mean = 161356 / 567 Arithmetic Mean = 284.5 Since 284.5 is between the given options (1) 284 and (2) 283.5, the correct answer is (4) None of these, which best represents this solution as the other options are not precise.

Key concepts

Arithmetic Mean Formula

The arithmetic mean, commonly known as the average, is a fundamental concept in statistics and mathematics, which helps in understanding the central tendency of a dataset. The arithmetic mean formula is given by the expression \( \text{Arithmetic Mean} = \frac{\text{Sum of all terms}}{\text{Number of terms}} \). For a set of natural numbers, this is equal to \( \frac{1 + 2 + 3 + \ldots + n}{n} \), where \( n \) represents the number of natural numbers in the set.

When the numbers are in a sequence, like the first \( n \) natural numbers, the formula simplifies the process of finding their mean without actually summing each individual number. Considering a large set of natural numbers, calculating the arithmetic mean can become very efficient by using formulas, as we can avoid manual computation, which is time-consuming and prone to errors.

Sum of Natural Numbers

The sum of natural numbers is a classical problem that has fascinated mathematicians for centuries. The formula to find the sum of the first \( n \) natural numbers is \( \frac{n(n + 1)}{2} \). This formula is derived from the series \( 1 + 2 + 3 + \ldots + n \) and is attributed to the famous mathematician Carl Friedrich Gauss.

For example, to calculate the sum of the first 567 natural numbers, we use the formula \( \text{Sum} = \frac{567(567 + 1)}{2} \) and get \( \text{Sum} = \frac{567 \times 568}{2} \) which results in 161,356. This simple yet powerful formula is a fundamental tool used in various fields, including computer science, statistics and finance, for computational efficiency.

Mathematical Problem Solving

Mathematical problem solving is a critical skill that encompasses understanding the problem, devising a plan, carrying it out, and then looking back to check and interpret the results. It's not just about finding the answer, but also about understanding why a particular method works and how it can be applied to other problems.

In the context of the arithmetic mean problem, the steps involved include using the known sum formula for natural numbers, performing arithmetic operations, and finally applying the arithmetic mean formula. The capability to disassemble complex problems into manageable parts is what often leads to a correct solution, as seen in our exercise example. Additionally, verifying your results against the given options ensures that mistakes and misunderstandings are caught early, teaching students the importance of reviewing their work.