Determining Arithmetic Sequence Properties from General Term

When working with general terms in mathematics, it's often necessary to determine if a sequence is arithmetic, meaning it follows a linear pattern. This guide will explore how to analyze the given general term to determine if a sequence is arithmetic. We'll use algebraic manipulation and calculation techniques to check for the presence of a common difference.

Introduction to Arithmetic Sequences

An arithmetic sequence is a sequence of numbers where the difference between any two successive members is constant. This constant difference is known as the common difference.

General Form of an Arithmetic Sequence

The general form of an arithmetic sequence is given by:

an Pn Q, where P is a polynomial of degree 1 (linear) and Q is a constant.

Any general term that does not follow this form is not an arithmetic sequence.

Example Analysis: an n^2 5n 6 / n^2

Let's analyze the given general term: ann2 5n 6n2. To determine if this forms an arithmetic sequence, we can simplify and check for a common difference.

Step 1: Simplify the General Term

First, simplify the numerator and denominator:

an n2 5n 6n2 1 5n 6n2

Clearly, the simplified expression is not linear, as it contains terms with nm for m 0. Therefore, this general term does not fit the form of an arithmetic sequence.

Step 2: Check for a Common Difference

To confirm, let's calculate the first few terms of the sequence:

an1 1 5 6 12

an2 2 5/2 6/22 7 5/2 6/4 7 2.5 1.5 11

an3 3 5/3 6/32 8.333 5/3 2/3 8.333 1.667 0.667 10.667

Now, let's check the differences:

an2 - an1 11 - 12 -1

an3 - an2 10.667 - 11 -0.333

-1 ≠ -0.333

Since the differences are not equal, the sequence is not arithmetic.

Another Example: an n^2 5n 6 / n^2

Let's consider another example: ann2 5n 6n2.

Step 1: Simplify the General Term

Simplify the expression:

an n 5/n 6/n2 n 5/n 6/n2

This form is not linear, so it is not an arithmetic sequence.

Step 2: Check for a Common Difference

Calculate the first few terms:

an1 1 5 6 12

an2 2 5/2 6/22 7 5/2 6/4 7 2.5 1.5 11

an3 3 5/3 6/32 8.333 5/3 2/3 8.333 1.667 0.667 10.667

Check the differences:

an2 - an1 11 - 12 -1

an3 - an2 10.667 - 11 -0.333

-1 ≠ -0.333

Therefore, this sequence is not arithmetic.

Conclusion

To determine if a sequence is arithmetic based on its general term, we must ensure the general term can be expressed in the form an Pn Q, where P is a linear polynomial and Q is a constant. We can use algebraic manipulation to simplify the general term and check if the differences between successive terms are constant. If the differences are not equal, the sequence is not arithmetic.