Understanding the Arithmetic Mean of a Sequence

The question asks for the arithmetic mean of the sequence 1, 8, 27, 64, ... up to n terms. First, let's identify the pattern in this sequence.

The terms are:

• 1st term = 1 = \(1^3\)
• 2nd term = 8 = \(2^3\)
• 3rd term = 27 = \(3^3\)
• 4th term = 64 = \(4^3\)

It is clear that the k-th term of the sequence is \(k^3\). Therefore, the sequence is the sequence of cubes of natural numbers, and the n-th term is \(n^3\).

The sequence up to n terms is 1, 8, 27, 64, ..., \(n^3\).

Calculating the Sum of the First n Terms

To find the arithmetic mean of these n terms, we need to calculate their sum and then divide by the number of terms, which is n.

The sum of the first n terms is the sum of the first n cubes:

Sum (S) = \(1^3 + 2^3 + 3^3 + \dots + n^3 = \sum_{k=1}^{n} k^3\)

There is a known formula for the sum of the first n cubes. The formula is:

\(\sum_{k=1}^{n} k^3 = \left( \frac{n(n+1)}{2} \right)^2 = \frac{n^2(n+1)^2}{4}\)

Finding the Arithmetic Mean

The arithmetic mean is defined as the sum of the terms divided by the number of terms. In this case, the number of terms is n.

Arithmetic Mean = \(\frac{\text{Sum of the first n terms}}{\text{Number of terms}}\)

Arithmetic Mean = \(\frac{\sum_{k=1}^{n} k^3}{n}\)

Substitute the formula for the sum of cubes into the arithmetic mean formula:

Arithmetic Mean = \(\frac{\frac{n^2(n+1)^2}{4}}{n}\)

Simplifying the Expression

Now, let's simplify the expression for the arithmetic mean:

Arithmetic Mean = \(\frac{n^2(n+1)^2}{4} \times \frac{1}{n}\)

Arithmetic Mean = \(\frac{n^2(n+1)^2}{4n}\)

We can cancel out one 'n' from the numerator and the denominator:

Arithmetic Mean = \(\frac{n \cdot n \cdot (n+1)^2}{4n}\)

Arithmetic Mean = \(\frac{n(n+1)^2}{4}\)

This is the formula for the arithmetic mean of the sequence 1, 8, 27, 64, ... up to n terms.

Comparing with Options

Let's compare the derived formula with the given options:

Option 1: \(\frac{{{\rm{n}}\left( {{\rm{n}} + 1} \right)}}{2}\)

Option 2: \(\frac{{{\rm{n}}{{\left( {{\rm{n}} + 1} \right)}^2}}}{2}\)

Option 3: \(\frac{{{\rm{n}}{{\left( {{\rm{n}} + 1} \right)}^2}}}{4}\)

Option 4: \(\frac{{{{\rm{n}}^2}{{\left( {{\rm{n}} + 1} \right)}^2}}}{4}\)

Our derived formula, \(\frac{n(n+1)^2}{4}\), matches Option 3.

Revision Table: Key Formulas

Additional Information: Understanding Arithmetic Mean and Series

The arithmetic mean is a fundamental concept in statistics and mathematics, representing the average value of a set of numbers. For a sequence or series, finding the arithmetic mean often requires first finding the sum of the terms.

The sequence 1, 8, 27, 64, ... is a sequence of perfect cubes. This type of sequence falls under the study of series and sequences, specifically power sums.

Understanding the formulas for sums of powers (like sum of first n integers, sum of first n squares, sum of first n cubes) is crucial for solving problems involving arithmetic means of such sequences. These formulas are derived using various methods, including mathematical induction.

In general, the arithmetic mean provides a single value that summarizes the central tendency of a dataset or sequence.