Problem 9 - Algebra Form and Function

Question

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

Step-by-Step Solution

Answer

Answer: The arithmetic mean of the given sequence is 2.

Step 1: Identify the sequence of numbers

The sequence of numbers is given as \(a_i = 2\) for \(i = 1, 2, \ldots, 10\). The sequence is: $$2, 2, 2, 2, 2, 2, 2, 2, 2, 2$$

Step 2: Calculate the sum of the sequence

Add all the elements in the sequence together: $$2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 = 20$$

Step 3: Divide the sum by the total number of elements

Now, we need to divide the sum of the sequence (20) by the total number of elements in the sequence (10): $$\bar{a} = \frac{20}{10} = 2$$ So, the arithmetic mean, or average, of the given sequence is \(\bar{a} = 2\).

Key Concepts

Sequence

In mathematics, a sequence is an ordered list of numbers. Each number in this list is called a term. Sequences can be finite or infinite, depending on the number of terms they contain.
Finite sequences have a specific number of elements, while infinite sequences continue indefinitely. In the original exercise, the sequence is given by \(a_i = 2\) for each \(i\) from 1 to 10. This means every term in the sequence is 2.
Thus, the sequence looks like this:

• 2, 2, 2, 2, 2, 2, 2, 2, 2, 2

Such sequences, where all terms are the same, are called constant sequences. Understanding the arrangement and properties of sequences is crucial for calculations involving concepts like the arithmetic mean.

Sum of a Sequence

The sum of a sequence is the result of adding all the terms of the sequence together. It is an important concept, especially when calculating averages or performing statistical analysis.
In the original exercise, each term of the sequence is 2 and there are a total of 10 terms. To find the sum, we simply add all these terms:

• \(2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 = 20\)

Calculating the sum of a sequence might be straightforward for constant sequences, but for sequences with varying terms, it requires adding each individual term carefully.
Knowing how to sum a sequence accurately is vital for deriving further statistics like the mean or median.

Number of Elements

The number of elements in a sequence is crucial when you need to calculate other statistics, such as the average or mean. It tells us how many terms are there in the sequence.
In the exercise provided, the number of elements is 10, since the sequence is specified from \(i = 1\) to \(i = 10\).
This count of elements helps in performing calculations like finding the arithmetic mean, where we divide the total sum by the number of elements. Knowing how to identify and interpret the number of elements is a basic yet essential skill in sequence analysis.