Question

Find \(\bar{a}\). $$ a_{i}=2 i, i=1, \ldots, 10 $$

Short answer

Answer: The average of the given arithmetic sequence is 11.

Step-by-step solution

Identify the Terms of the Sequence

For this arithmetic sequence, we have a general term \(a_i = 2i\), where \(i\) ranges from 1 to 10. Let's find the terms of the sequence: \(a_1 = 2(1) = 2\) \(a_2 = 2(2) = 4\) \(a_3 = 2(3) = 6\) \(a_4 = 2(4) = 8\) \(a_5 = 2(5) = 10\) \(a_6 = 2(6) = 12\) \(a_7 = 2(7) = 14\) \(a_8 = 2(8) = 16\) \(a_9 = 2(9) = 18\) \(a_{10} = 2(10) = 20\)

Find the Sum of the Terms

Now that we have all the terms of the sequence, let's find their sum: \(S = a_1 + a_2 + a_3 + \cdots + a_{10} = 2 + 4 + 6 + \cdots + 20\) Notice that this is an arithmetic series with a common difference of 2 between each term. We can use the arithmetic series formula to find the sum: \(S = n\frac{(a_1 + a_n)}{2}\) where \(n=10\) (the number of terms), \(a_1 = 2\) (the first term), and \(a_n = 20\) (the last term) \(S = 10\frac{(2 + 20)}{2}\)

Calculate the Sum

Now let's calculate the sum: \(S = 10\frac{(2 + 20)}{2} = 10\frac{22}{2} = 10(11) = 110\)

Find the Average

Finally, let's find the average (arithmetic mean) of the sequence by dividing the sum by the number of terms: \(\bar{a} = \frac{S}{n} = \frac{110}{10} = 11\) So, the average of the given arithmetic sequence is \(\bar{a} = 11\).

Key concepts

Sequence

An arithmetic sequence is a list of numbers where each term after the first is obtained by adding a constant difference to the previous term. For instance, in the given exercise the formula \(a_i = 2i\) defines an arithmetic sequence that starts from \(i = 1\) and goes up to \(i = 10\). The numbers you receive are: 2, 4, 6, 8, 10, 12, 14, 16, 18, and 20. These numbers are called the terms of the sequence.
To create this type of sequence, identify the first term and the consistent value added, which is called the common difference. Here, the first term \(a_1\) is 2 and the common difference is 2 as well. Other sequences may have positive or negative differences, affecting how the sequence increases or decreases over its terms.

Average

In mathematics, the average is a number used to summarize a set of values, which provides insight into the general size of the numbers. For sequences, determining the average helps reveal the central tendency—the middle or typical number in the set.
In the exercise, calculating the average of the sequence involves first finding the sum of all terms, which totals 110. Then this sum is divided by the number of terms, which is 10. Hence, the average (also known as the mean) of this sequence results in \(\bar{a} = 11\).
The process of averaging is crucial in various statistical and mathematical contexts because it supplies a simple value for comparison and analysis.

Sum

The sum of a sequence is the result of adding all its terms together. In the context of arithmetic sequences, this sum can be calculated quickly using a formula. Instead of adding each term like \(2 + 4 + 6 + \dots + 20\) separately, we utilize the arithmetic series formula:

• \(S = n\frac{(a_1 + a_n)}{2}\)

Here, \(n\) is the number of terms, \(a_1\) is the first term, and \(a_n\) is the last term. Using this formula simplifies the process significantly as seen in the solution where the total sum is calculated as 110.
This method is efficient, especially for longer sequences, ensuring all terms are accounted for correctly and quickly.

Arithmetic Mean

The arithmetic mean is another term for what is commonly known as the average. It involves taking the sum of a set of terms and dividing it by the number of terms. This method gives a central value that represents the entire set.
In the exercise, after calculating the total sum of 110 from the sequence, the arithmetic mean is found by dividing by 10—the number of terms. As a result, the mean of this sequence is 11.
This concept of arithmetic mean is fundamental across various fields such as economics, engineering, and everyday problem-solving, providing a basic yet powerful way to analyze data sets.