Problem 14 - Precalculus with Limits

Question

In Exercises 5 - 14, determine whether the sequence is arithmetic. If so, find the common difference. \( 1^2, 2^2, 3^2, 4^2, 5^2, \cdots \)

Step-by-Step Solution

Answer

No, the provided sequence is not an arithmetic sequence.

Step 1: Understanding the Sequence

Firstly, it's understood that the sequence provided is the square of consecutive natural numbers. It is actually presented as: 1, 4, 9, 16, 25, \cdots.

Step 2: Checking the Common Difference

In order to identify whether the sequence is arithmetic, it's essential to find the differences between successive terms. In this case, the differences are: \(4 - 1 = 3\), \(9 - 4 = 5\), \(16 - 9 = 7\), \(25 - 16 = 9\), and so on.

Step 3: Concluding the Sequence Type

By verifying the calculation from Step 2, it's clear that the difference between consecutive terms is not constant. This implies that the given sequence cannot be considered an arithmetic sequence.

Key Concepts

Common Difference

In an arithmetic sequence, the common difference is the key factor that allows a sequence to be classified as arithmetic. The common difference is the constant value that we add or subtract from one term to get to the next. Determining whether a sequence has a common difference involves checking each pair of consecutive terms in the sequence.
For example, in the arithmetic sequence 3, 6, 9, 12, you can see that each term increases by 3. Here, 3 is the common difference.

• It is calculated by subtracting the first term from the second term.
• This process is repeated to ensure that the difference remains consistent throughout the sequence.

If the difference is consistent, we can confirm that the sequence is arithmetic. However, if the differences between consecutive terms are not equal, like in the sequence of squared numbers (1, 4, 9, 16, 25), that means there's no common difference, and it's not an arithmetic sequence.

Natural Numbers

The sequence given in the exercise is based on the squares of natural numbers. But what exactly are natural numbers? Natural numbers are the set of positive integers starting from 1. They are sometimes called counting numbers.
In mathematics, natural numbers are often denoted by

• Natural numbers: \(1, 2, 3, 4, 5, \ldots\)
• Square of natural numbers: \(1^2, 2^2, 3^2, 4^2, 5^2, \ldots\)

It is important to note that when you square natural numbers, the result is not necessarily an arithmetic sequence due to the increasing non-uniform differences between terms. Understanding the properties of natural numbers can help in identifying their sequences.

Sequence Analysis

Sequence analysis involves examining sequences to understand their properties and relationships. It can involve checking sequences for specific patterns, such as being arithmetic or geometric.
To determine if a sequence is arithmetic, sequence analysis often involves:

• Checking the difference between consecutive terms (finding the common difference).
• Verification through repeated calculations between consecutive terms.

In the case of the sequence of squared numbers (1, 4, 9, 16, 25), a deeper sequence analysis reveals that the differences between consecutive terms (3, 5, 7, 9) do not form a constant difference. This shows that the pattern is quadratic rather than arithmetic. Sequence analysis helps in identifying such distinct characteristics to properly categorize sequences and solve problems related to them.