Question

The mean of the series \(a, a+d, a+2 d \ldots \ldots . a+(2 n+1) d\) is (a) \(a+[(2 n+1) / 2] d\) (b) \(a+(n+1) d\) (c) \(a+(2 n+1) d\) (d) \(a+[(2 n-1) / 2] d\)

Short answer

The mean of the given series is \(a + \frac{(2n+1)d}{2}\).

Step-by-step solution

Calculate the sum of the series

We have the series: \(a, a+d, a+2d, \ldots, a+(2n+1)d\). We can see that this is an arithmetic series with common difference d. The sum of an arithmetic series can be calculated using the formula: \[S = \frac{n}{2} (a_1 + a_n)\] where n is the number of terms, \(a_1\) is the first term and \(a_n\) is the last term. In our case, \(a_1 = a\) and \(a_n = a + (2n+1)d\). The total number of terms in the series is (2n+2). Now, let's find the sum of the series, S: \[S = \frac{2n+2}{2} (a + a + (2n+1)d)\]

Simplify the sum formula

Upon simplifying the formula for the sum, we get: \[S = (n+1)(2a+2n d + d)\] \[S = (n+1)(2a+(2n+1)d)\]

Calculate the mean of the series

Now we will find the mean of the series, which is the sum of the series divided by the total number of terms. As mentioned earlier, the total number of terms is (2n+2). Mean = \(\frac{S}{2n+2}\) Substitute the formula for S from Step 2: \[Mean = \frac{(n+1)(2a+(2n+1)d)}{2n+2}\]

Simplify the mean formula

Simplify the mean formula: \[Mean = \frac{(n+1)(2a+(2n+1)d)}{(n+1)(2)}\] \[Mean = \frac{2a+(2n+1)d}{2}\] Our final expression for the mean is: \[Mean = a + \frac{(2n+1)d}{2}\] Now we can see that this matches option (a): \(a + \frac{(2n+1)d}{2}\). So, the correct answer is (a).

Key concepts

Arithmetic Mean

The arithmetic mean is a measure of central tendency, similar to the average. In the context of a sequence, it represents the average value of all the terms in the sequence. For an arithmetic progression, the mean is calculated by dividing the sum of all terms by the number of terms.

In our exercise, the series under consideration was an arithmetic series with terms defined as follows: \(a, a+d, a+2d, \ldots, a+(2n+1)d\). To find its mean, we first needed to calculate the sum of the entire series—this sum was then divided by the number of terms in the sequence.

Typically, the formula for the mean \(M\) of an arithmetic series is given as:

• \( M = \frac{S}{n}\)

where \(S\) is the sum of the series and \(n\) is the number of terms. Understanding the arithmetic mean helps identify the center of the data set and provides insight into the progression's overall trend.

Sum of Series

The sum of a series refers to the total when all the terms in the sequence are added together. For an arithmetic series, this sum can be easily calculated using a specific formula. In arithmetic sequences, where each term increases by a constant difference, this formula is very useful.

The sum \(S\) of an arithmetic series is given by:

• \( S = \frac{n}{2} \times (a_1 + a_n) \)

where \(a_1\) is the first term, \(a_n\) is the last term, and \(n\) is the number of terms in the series.

In our example, the series was \(a, a+d, a+2d, \ldots, a+(2n+1)d\), and using the formula, we managed to simplify it to \( S = (n+1)(2a+(2n+1)d) \). Calculating the sum is the foundational step before finding the mean and other properties of the series.

Number of Terms

The number of terms in a sequence is a fundamental concept used to understand the length and extent of the sequence. In our particular arithmetic series, determining the number of terms was crucial as it directly influenced both the sum and the mean calculations.

The series given was \(a, a+d, a+2d, \ldots, a+(2n+1)d\). An important point to note in arithmetic sequences is that the number of terms \(n\) can be found by counting all terms from the first to the last term, inclusive. In this case, the number of terms was given as \(2n + 2\).

Understanding the sequence's length helps calculate its sum correctly, ensuring that any subsequent operations, like finding the mean, are accurate. It's essential for students to identify how to quickly determine the number of terms in any given progression.

Arithmetic Progression Formula

The arithmetic progression formula is essential for evaluating and working with sequences where each term differs from the previous one by a fixed, constant amount. This difference is known as the common difference \(d\).

The general formula for the \(n\)-th term in an arithmetic sequence is:

• \( a_n = a_1 + (n-1) \cdot d \)

where \(a_1\) is the first term and \(d\) is the common difference.

In our example, the sequence is defined by terms \(a, a+d, a+2d, \ldots, a+(2n+1)d\). Using this formula allows us not only to define each term explicitly but also to systematically derive other fundamental properties of the series, like its sum or mean.

Mastery of the arithmetic progression formula is critical for solving a wide range of mathematical problems, as it provides a logical foundation for manipulating and understanding linear sequences.