Problem 17 - Algebra Form and Function

Question

Find the mean of \(1,2,3, \ldots, n,\) if \(n\) is (a) 3 (b) 4 (c) 5 (d) 6 (e) \(2 \mathrm{k}\) (f) \(2 \mathrm{k}+1\)

Step-by-Step Solution

Answer

Question: Determine the mean of the numbers from 1 to n for the following cases: (a) n=3, (b) n=4, (c) n=5, (d) n=6, (e) n=2k, and (f) n=2k+1. Answer: (a) For n=3, the mean is 2. (b) For n=4, the mean is 2.5. (c) For n=5, the mean is 3. (d) For n=6, the mean is 3.5. (e) For n=2k, the mean is \(\frac{1}{2}(2k+1)\). (f) For n=2k+1, the mean is \(k+1\).

Step 1: Find the sum of the numbers for each case

For each case, we have to find the sum of numbers from 1 to n. (a) n=3: Sum is \(1+2+3=6\) (b) n=4: Sum is \(1+2+3+4=10\) (c) n=5: Sum is \(1+2+3+4+5=15\) (d) n=6: Sum is \(1+2+3+4+5+6=21\) For (e) and (f), we can use the formula for the sum of the first n natural numbers, which is: Sum = \(\frac{n(n+1)}{2}\) (e) n=2k: Sum is \(\frac{2k(2k+1)}{2}=k(2k+1)\) (f) n=2k+1: Sum is \(\frac{(2k+1)(2k+1+1)}{2}=\frac{(2k+1)(2k+2)}{2}=(2k+1)(k+1)\)

Step 2: Find the mean for each case

Divide the sum by the total number of values (n) to find the mean. (a) Mean when n=3 is \(\frac{6}{3}=2\) (b) Mean when n=4 is \(\frac{10}{4}=2.5\) (c) Mean when n=5 is \(\frac{15}{5}=3\) (d) Mean when n=6 is \(\frac{21}{6}=3.5\) (e) Mean when n=2k is \(\frac{k(2k+1)}{2k}=\frac{1}{2}(2k+1)\) (f) Mean when n=2k+1 is \(\frac{(2k+1)(k+1)}{2k+1}=k+1\) So, the mean of the numbers from 1 to n is as follows: (a) For n=3, the mean is 2. (b) For n=4, the mean is 2.5. (c) For n=5, the mean is 3. (d) For n=6, the mean is 3.5. (e) For n=2k, the mean is \(\frac{1}{2}(2k+1)\). (f) For n=2k+1, the mean is \(k+1\).

Key Concepts

Sum of Natural Numbers

The concept of the sum of natural numbers is intriguing and foundational in understanding sequences and series. Natural numbers are the set of positive integers starting from 1, progressing as 1, 2, 3, and so on. To calculate the sum of these numbers up to any given number, say \( n \), we use a simple formula. The formula to find the sum of the first \( n \) natural numbers is given by:\[\text{Sum} = \frac{n(n+1)}{2}\]This formula works due to the consistent step-up in value between consecutive numbers, which means that pairing terms from the beginning and end consistently gives the same total. Here's how the formula can be proven through a simple example:

• Consider \( n = 4 \), the natural numbers are 1, 2, 3, 4.
• The sum can be calculated as \(1 + 2 + 3 + 4 = 10\).
• Using the formula: \( \frac{4(4+1)}{2} = \frac{20}{2} = 10 \).

The easy calculation shows how using the formula quickly provides the solution without individually adding each number, revealing the elegance of mathematic equations.

Arithmetic Progression

An arithmetic progression (AP) is a sequence of numbers in which the difference between consecutive terms is constant. This difference is known as the common difference. For example, the sequence 1, 2, 3, 4, 5 is an arithmetic progression where the common difference is 1. Similarly, the sequence 3, 7, 11, 15 is an AP with a common difference of 4.

The general formula for finding the \( n \)-th term of an arithmetic progression where the first term is \( a \) and the common difference is \( d \) is:\[a_n = a + (n-1)d\]For example, in the sequence 2, 4, 6, 8, where \( a = 2 \) and \( d = 2 \):

• The third term would be \( 2 + (3-1) \cdot 2 = 6 \).
• This matches the third term in the given sequence.

The sum of an arithmetic progression can also be calculated using the sum formula of AP:\[\text{Sum} = \frac{n}{2} \times (\text{First Term} + \text{Last Term})\]Understanding APs is vital for calculating and understanding more complex patterns in mathematics, as they often form the basis for more advanced sequences.

Generalized Mean Calculation

The concept of mean is integral in statistics and represents the average of a set of values. In generalized mean calculation, the focus is on finding the mean or average of a particular sequence of numbers, effectively summarizing them with a single value.

To find the mean of any sequence, you first sum up the numbers in the sequence and then divide by the number of values. This can be expressed as:\[\text{Mean} = \frac{\text{Sum of all values}}{\text{Total number of values}}\]For example, to find the mean of the numbers 1, 2, 3, and 4:

• First, calculate the sum: \( 1 + 2 + 3 + 4 = 10 \).
• Next, divide by the total number of values (4): \( \frac{10}{4} = 2.5 \).

This principle applies whether you are finding the mean for a fixed set of numbers or more abstract representations like the arithmetic progression or other series where general formulas for the sum can simplify the calculations.