Question

The arithmetic mean of the series \(1,3,3^{2}, \ldots \ldots 3^{\mathrm{n}-1}\) is (1) \(\frac{3^{n}}{2 n}\) (2) \(\frac{3^{\mathrm{n}}-1}{2 \mathrm{n}}\) (3) \(\frac{3^{\mathrm{n}-1}}{\mathrm{n}+1}\) (4) None of these

Short answer

Answer: \(\frac{3^n - 1}{2n}\)

Step-by-step solution

Finding the sum of the series

The given series is a geometric series with the first term \(a = 1\) and the common ratio \(r = 3\). The sum of the first n terms of a geometric series is given by: $$S_n = \frac{a(r^n - 1)}{r - 1}$$ In our case, \(a = 1\), \(r = 3\), so we have: $$S_n = \frac{1(3^n - 1)}{3 - 1}$$

Finding the arithmetic mean

The arithmetic mean of a series is the sum of the terms divided by the number of terms. In this case, the number of terms is n. So the arithmetic mean (A) is given by: $$A = \frac{S_n}{n}$$ Substituting the value of \(S_n\) from step 1, we get: $$A = \frac{(3^n - 1)}{2n}$$

Comparing our result with the given options

Now, let's compare our result with the given options: Option (1): \(\frac{3^n}{2n}\) Option (2): \(\frac{3^n - 1}{2n}\) Option (3): \(\frac{3^{n-1}}{n+1}\) Option (4): None of these Our result matches with option (2). Hence, the arithmetic mean of the given series is \(\frac{3^n - 1}{2n}\).

Key concepts

Geometric Series

A geometric series is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. In our series, we start with the number 1, and each subsequent term is multiplied by 3. This makes our common ratio 3. So, the sequence looks like this: 1, 3, 9, 27, and so on, up to the nth term written as \(3^{n-1}\).

Understanding geometric series is useful because it allows us to quickly find the sum of the terms up to a desired point. This is particularly helpful in solving problems where knowing the total value of a patterned sequence is required. In this exercise, the total sum of the terms was found using the formula:

• \(S_n = \frac{a(r^n - 1)}{r - 1}\)

where \(a=1\) is the first term, and \(r=3\) is the common ratio. By plugging these into the formula, you can achieve the sum \(S_n\) of the first n terms.

Sum of Series

The concept of summing a series involves adding up all terms in the sequence. For a geometric series, it's especially useful that we have a specific formula to calculate this sum, which is:

• \(S_n = \frac{a(r^n - 1)}{r - 1}\)

This formula allows you to calculate the sum without the need to add each term individually, which is handy for long sequences.

In our particular example, the series is \(1, 3, 3^2, \, ... \, 3^{n-1}\). By applying the formula, we set \(a = 1\) (the first term), \(r = 3\), and we substitute these to find \(S_n = \frac{3^n - 1}{2}\). This formula works because it accounts for both the growth factor (the common ratio) and the number of terms you're adding together.

Arithmetic Mean Formula

The arithmetic mean is a way to find the average value of a list of numbers. To find the arithmetic mean, you simply add up all the numbers and then divide by how many numbers there are. It's a measure of central tendency, which helps us understand the average behavior of the series.

In a geometric series, like our exercise, you can find the arithmetic mean of the terms by dividing the sum of all terms (\(S_n\)) by the number of terms (n). For our specific series, we found:

• \(A = \frac{S_n}{n} = \frac{3^n - 1}{2n}\)

This gives us the average value for the entire sequence. In the exercise, the solution used this formula to find the arithmetic mean, which matched with option (2) in the given choices. Understanding how to use this formula is crucial for calculating averages effectively. It connects the concept of series and summation to real-world applications where averages are frequently used.