Problem 7 - Algebra Form and Function

Question

Find \(\bar{a}\). $$ a_{i}=i^{2}, i=1, \ldots, 6 $$

Step-by-Step Solution

Answer

Answer: The mean of the sequence is \(\frac{91}{6}\).

Step 1: Identify the sequence of numbers

The given sequence is \(a_{i} = i^2\) for \(i = 1, \ldots, 6\). So, the sequence of numbers is: $$ 1^2, 2^2, 3^2, 4^2, 5^2, 6^2 $$

Step 2: Find the sum of the sequence

Now, we will find the sum of all terms in the sequence: $$ 1^2 + 2^2 + 3^2 + 4^2 + 5^2 + 6^2 = 1 + 4 + 9 + 16 + 25 + 36 = 91 $$

Step 3: Calculate the mean

Finally, we will find the mean \(\bar{a}\) by dividing the sum of the terms by the total number of terms: $$ \bar{a} = \frac{91}{6} $$ So, the mean of the sequence is \(\frac{91}{6}\).

Key Concepts

Sequence

In mathematics, a sequence is simply a list of numbers arranged in a particular order. Each number in the sequence is referred to as a term. When working with sequences, it's important to note the pattern or rule that defines the arrangement of the terms. For instance, in our exercise, the sequence is defined by the rule: \(a_i = i^2\). This means each term is the square of its position number. Thus, the terms in our sequence for \(i = 1, \ldots, 6\) are:

• \(1^2\) which is 1
• \(2^2\) which is 4
• \(3^2\) which is 9
• \(4^2\) which is 16
• \(5^2\) which is 25
• \(6^2\) which is 36

Understanding how to identify patterns in sequences can help in predicting future terms or computing specific values within the sequence.

Sum of Squares

The 'sum of squares' in this context refers to the result of adding up squared numbers from the defined sequence. In essence, when we take each number in our sequence and square it, the resultant numbers are subsequently added together to find their sum. This is a common requirement in mathematical problems involving data sets. Let's break down the sum of squares from the exercise:

• Add the squared terms: \(1 + 4 + 9 + 16 + 25 + 36\)
• Each term originates from squaring a number in the sequence; hence it's called a 'sum of squares.'

Resulting in a total sum of 91. This process is crucial as it helps determine not only the arithmetic mean but also in statistical analysis such as variance calculations when applied to larger data sets.

Mean Calculation

The arithmetic mean is essentially the average of a list of numbers. To find it, you sum up all the values and then divide by the number of values. This measure provides insight into the central tendency of the numbers. It's a fundamental concept in statistics and a crucial part of understanding data sets.In the given exercise, following sequence identification and sum of squares, the final task is to compute the mean:

• The sum of all squared terms is 91.
• There are 6 numbers in the list.
• Divide the total sum by the count of numbers: \(\bar{a} = \frac{91}{6}\)

The result, which is \(\bar{a} = 15.1667\) (approximately), represents the average of the squared numbers. Understanding mean calculation helps in various fields such as finance, economics, and everyday decision-making.