Question

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

Short answer

Answer: The arithmetic mean of the sequence is 12.4.

Step-by-step solution

Write out the values of the sequence

For each \(i\) from 1 to 5, calculate the value of \(a_i=2^i\): 1. \(a_1 = 2^1 = 2\) 2. \(a_2 = 2^2 = 4\) 3. \(a_3 = 2^3 = 8\) 4. \(a_4 = 2^4 = 16\) 5. \(a_5 = 2^5 = 32\) The sequence is \(2, 4, 8, 16, 32\).

Calculate the sum of the sequence

Add together all the values of the sequence: \(2 + 4 + 8 + 16 + 32 = 62\).

Find the arithmetic mean \(\bar{a}\)

Divide the sum of the sequence by the number of values (5): \(\bar{a} = \frac{62}{5} = \frac{124}{10} = 12.4\). The arithmetic mean \(\bar{a}\) of the given sequence is 12.4.

Key concepts

Sequence

A sequence is a set of numbers arranged in a specific order, governed by a particular rule or formula. In our exercise, the sequence is generated by the rule \(a_i = 2^i\), where \(i\) is an integer starting from 1 and ending at 5. This means each term is determined by raising 2 to the power of the term's position in the sequence:

• When \(i = 1\), \(a_1 = 2^1 = 2\).
• When \(i = 2\), \(a_2 = 2^2 = 4\).
• And so on, up to \(i = 5\), \(a_5 = 2^5 = 32\).

The entire sequence here is \(2, 4, 8, 16, 32\). Understanding sequences is crucial as they form the basis for more complex mathematical concepts like series, functions, and progressions.

Exponential Growth

Exponential growth is a pattern of data that shows greater increases with passing time, creating the shape of an exponential curve. In our sequence, \(a_i = 2^i\), each term is exponentially larger than the previous one. This happens because 2 is raised to the increasing power of \(i\).

• The growth from 2 to 4 is a 100% increase.
• The growth from 4 to 8 is also a 100% increase, and this pattern continues.

The hallmark of exponential growth is its constant ratio of change, unlike linear growth, which increases by a constant amount. As you progress through the sequence, the values multiply rather than add, demonstrating how powerful exponential functions can be in modeling rapid changes.

Summation

Summation is the process of adding a sequence of numbers to find their total. In mathematical notation, it is often represented using the sigma symbol \(\Sigma\). Here, we are tasked to find the total of our sequence: \(2, 4, 8, 16, 32\).

• First, write out all the terms: \(2 + 4 + 8 + 16 + 32\).
• Then proceed to add them step-by-step: \(2 + 4 = 6\), \(6 + 8 = 14\), \(14 + 16 = 30\), and so forth until you reach 62.

The final sum of 62 helps us in further calculations, like finding the mean. Summation is a fundamental operation in mathematics, providing a foundation for calculus and numerical analysis.

Step-by-Step Solution

Step-by-step solutions break down complex problems into manageable parts, making them easier to understand and solve. In solving for the arithmetic mean of our sequence:
1. We first wrote out each term using the formula \(a_i = 2^i\) to understand the values we are dealing with. This gives a clearer picture of the numbers involved.2. Then, we calculated the sum of the sequence numerically to ensure accuracy and correctness.3. Finally, to find the arithmetic mean, we divided the total sum by the number of terms, which provided direct insight into the average size of these exponential growth values.
With each step clearly outlined, it helps students grasp how to handle the process incrementally, strengthening their problem-solving skills and ensuring the accuracy of their work.