Question

Find the arithmetic mean of the series \(1,3,5, \ldots \ldots \ldots(2 \mathrm{n}-1) .\) (1) \(\frac{2 \mathrm{n}-1}{\mathrm{n}}\) (2) \(\frac{2 \mathrm{n}+1}{\mathrm{n}}\) (3) \(\mathrm{n}\) (4) \(\mathrm{n}+2\)

Short answer

Answer: The arithmetic mean of the given series is \(n\).

Step-by-step solution

Identifying the series type and main components

The given series is an arithmetic series with the first term, a=1 and common difference d=2. The number of terms in the series is n. The general term of the series can be written as \(t_n = 1 + (n-1)\cdot 2\) which simplifies to \(t_n = 2n-1\).

Sum of the arithmetic series

The sum of an arithmetic series can be calculated using the formula: \(S_n = \frac{n}{2}\cdot(a_1 + a_n)\). In our case, \(S_n = \frac{n}{2}\cdot(1 + (2n-1))\).

Simplify the sum formula

Now, let's simplify the sum formula. On simplifying the sum formula, we get: \(S_n = \frac{n}{2}\cdot(1+2n-1) = \frac{n}{2}\cdot(2n)\).

Calculate the arithmetic mean

The arithmetic mean is calculated by dividing the sum of the series by the number of terms, which is n in our case. Hence, the arithmetic mean can be written as: \(Mean = \frac{S_n}{n}\). Substitute the value of \(S_n\) in the formula, we get: \(Mean = \frac{\frac{n}{2}\cdot(2n)}{n}\).

Simplify the expression for the arithmetic mean

Let's simplify the expression for the arithmetic mean by cancelling out the common factors. We can cancel out n, resulting in: \(Mean = n\). The arithmetic mean of the given series is \(n\), so option (3) is the correct answer.

Key concepts

Arithmetic Series

An arithmetic series is a type of sequence where the difference between terms remains constant. This consistent value is known as the common difference. In the exercise, the series

• begins with the first term 1
• has a common difference of 2

This means each term is determined by adding 2 to the previous term. For instance, the series starts at 1, then goes to 3 (1 + 2), next is 5 (3 + 2), and so forth. This predictable pattern defines the arithmetic series.

Sum of Arithmetic Series

Calculating the sum of an arithmetic series involves using a specific formula. The sum, denoted as \(S_n\), can be found with \[S_n = \frac{n}{2} \cdot (a + l),\]where:

• \(n\) is the number of terms
• \(a\) is the first term
• \(l\) is the last term

For our series, the formula becomes \[S_n = \frac{n}{2} \cdot (1 + (2n-1)),\]which simplifies to \[S_n = n^2.\]This formula makes it straightforward to calculate the total of all terms when the first and last terms are known.

Arithmetic Progression

An arithmetic progression is essentially the framework of how an arithmetic series is structured. Each term within the progression is created by adding a constant value, called the common difference, to the previous term. In the given series example, the progression starts with 1 and follows this rule: each subsequent number is the previous number plus 2. This forms the pattern: 1, 3, 5, and so on. This uniform step size is what characterizes the sequence as an arithmetic progression.

Mean Calculation

The arithmetic mean is the average of a set of numbers. It's a measure to find the central value. To find the arithmetic mean of a series, you divide the sum of the terms by the number of terms. For the given series, after determining \(S_n\) is \[S_n = n^2,\]we calculate the mean by \[Mean = \frac{S_n}{n} = \frac{n^2}{n} = n.\]This result simplifies the central tendency of the series to be equal to \(n\). It shows that when you average all the terms in this arithmetic series, the value coincidently equates to the number of terms itself.