Calculating the Mean of a Geometric Series

The problem asks us to find the mean of the series 1, 2, 4, 8, ..., 2ⁿ.

Understanding the Given Series

Let's examine the terms of the series:

• The first term is 1.
• The second term is 2.
• The third term is 4.
• The fourth term is 8.
• ...
• The last term is 2ⁿ.

We can observe a pattern: each term is obtained by multiplying the previous term by 2. This type of series is known as a geometric series.

Identifying the Key Components of the Geometric Series

To find the mean, we need the sum of the terms and the number of terms.

• The first term, often denoted as 'a', is 1.
• The common ratio, often denoted as 'r', is the ratio of any term to its preceding term (e.g., 2/1 = 2, 4/2 = 2). So, r = 2.
• The terms can be written as 2⁰, 2¹, 2², 2³, ..., 2ⁿ.
• The exponents range from 0 to n. This means there are \(n - 0 + 1 = n + 1\) terms in the series.

So, we have a geometric series with:

• First term (a) = 1
• Common ratio (r) = 2
• Number of terms (N) = n + 1

Calculating the Sum of the Series

The sum (S) of a geometric series with first term 'a', common ratio 'r', and N terms is given by the formula:

\(S = \frac{a(r^N - 1)}{r - 1}\)

In our case, a = 1, r = 2, and N = n + 1. Substituting these values into the formula:

\(S = \frac{1(2^{n+1} - 1)}{2 - 1}\)

\(S = \frac{2^{n+1} - 1}{1}\)

\(S = 2^{n+1} - 1\)

The sum of the series 1, 2, 4, ..., 2ⁿ is \(2^{n+1} - 1\).

Calculating the Mean of the Series

The mean of a series is the sum of the terms divided by the number of terms.

\(\text{Mean} = \frac{\text{Sum of terms}}{\text{Number of terms}}\)

We found the sum of terms to be \(2^{n+1} - 1\), and the number of terms is \(n + 1\).

\(\text{Mean} = \frac{2^{n+1} - 1}{n + 1}\)

Comparing with the Options

Let's look at the given options:

1. \(\rm \frac{2^{n-1}-1}{n+1}\)
2. \(\rm \frac{2^{n+1}-1}{n}\)
3. \(\rm \frac{2^{n+1}-1}{n+1}\)
4. \(\rm \frac{2^{n-1}}{n}\)

Our calculated mean, \(\frac{2^{n+1} - 1}{n + 1}\), matches option 3.

Conclusion

The mean of the series 1, 2, 4, 8, ..., 2ⁿ is \(\frac{2^{n+1} - 1}{n + 1}\).

Revision Table: Key Concepts

Additional Information: Exploring Series and Mean

Understanding different types of series and how to calculate their sum and mean is fundamental in mathematics and statistics.

• Arithmetic Series: In contrast to a geometric series, an arithmetic series has a constant difference between consecutive terms (e.g., 3, 5, 7, 9...). The sum of an arithmetic series is given by \(S_N = \frac{N}{2}(2a + (N-1)d)\), where 'a' is the first term and 'd' is the common difference.
• Relationship between Sum and Mean: The formula for the mean directly depends on the sum and the number of terms. If you can calculate the sum and count the terms, you can find the mean.
• Applications: Geometric series appear in various fields, including finance (compound interest), population growth, and physics (radioactive decay). Calculating their mean helps understand the average value over a period or across terms.

Always identify the type of series (arithmetic, geometric, etc.) before applying summation formulas. Correctly determining the number of terms is also crucial for both sum and mean calculations.