Problem 11 - Algebra Form and Function

Question

Find \(\bar{a}\). $$ a_{i}=i / 2 \text { and } i=1, \ldots, 5 $$

Step-by-Step Solution

Answer

Answer: The mean of the sequence is \(\frac{3}{2}\).

Step 1: Write out the terms of the sequence

To find the terms of the sequence, we simply substitute each value of \(i\) into \(a_i = \frac{i}{2}\). This gives us: $$ a_1 = \frac{1}{2}, a_2 = \frac{2}{2}, a_3 = \frac{3}{2}, a_4 = \frac{4}{2}, a_5 = \frac{5}{2} $$

Step 2: Calculate the sum of the sequence

Now, we need to find their sum: $$ S = a_1 + a_2 + a_3 + a_4 + a_5 = \frac{1}{2} + \frac{2}{2} + \frac{3}{2} + \frac{4}{2} + \frac{5}{2} $$ Combining the terms, we get: $$ S = \frac{1+2+3+4+5}{2} = \frac{15}{2} $$

Step 3: Find the mean of the sequence

Finally, we find the mean by dividing the sum of the sequence by the number of terms. Since there are 5 terms in the sequence, we get: $$ \bar{a} = \frac{S}{n} = \frac{\frac{15}{2}}{5} = \frac{15}{2} \cdot \frac{1}{5} $$ Simplifying, we find: $$ \bar{a} = \frac{15}{10} = \frac{3}{2} $$ So, the mean of the sequence \(\bar{a}\) is \(\frac{3}{2}\).

Key Concepts

Arithmetic Sequence

An arithmetic sequence is a collection of numbers in which the difference between consecutive terms is constant. In simpler terms, if you take any term in the sequence and subtract its previous term, you will always get the same number. This difference is known as the "common difference."
For example, consider the sequence given in the exercise:

• The terms are expressed through the formula: \(a_i = \frac{i}{2}\).
• As you replace \(i\) with sequential integers from 1 to 5, you get the terms: \(\frac{1}{2}, 1, \frac{3}{2}, 2, \frac{5}{2}\).

The common difference here is \(\frac{1}{2}\), which is consistent across terms because each term increases by \(\frac{1}{2}\) as the sequence progresses.

Sequence Summation

Sequence summation is the process of adding all the terms in a sequence to form a single total. It's an essential concept, especially when calculating the mean of a sequence.
For the sequence in the exercise, the terms are

• \(\frac{1}{2}\), \(1\), \(\frac{3}{2}\), \(2\), \(\frac{5}{2}\).
• The sum of these terms can be simply calculated as:\[ S = \frac{1}{2} + 1 + \frac{3}{2} + 2 + \frac{5}{2} \].

By converting these fractions to have a common denominator and then adding them all up, we end up with: \(S = \frac{15}{2}\). This sum is crucial for the next step of finding the average or mean.

Average of Sequence

The average (or mean) of a sequence is a measure of central tendency that tells us the "central" or "typical" value of the set of numbers. The mean is found by dividing the total sum of all terms by the number of terms in the sequence.
For our sequence, the sum was calculated to be \(\frac{15}{2}\).

• The sequence consists of 5 terms: \(\frac{1}{2}, 1, \frac{3}{2}, 2, \frac{5}{2}\).
• To find the mean, you divide the sum by the total number of terms, which is 5:\[ \bar{a} = \frac{\frac{15}{2}}{5} \].

Performing this division gives us the mean as \(\frac{3}{2}\). This value tells us the typical value of the sequence.