Is the sequence 900, 450, 225 an arithmetic or geometric sequence?

It is a geometric sequence because each term is multiplied by a common ratio, specifically multiplied by 0.5.

What is the common ratio of the sequence 900, 450, 225?

The common ratio is 0.5, since 450 ÷ 900 = 0.5 and 225 ÷ 450 = 0.5.

Write the explicit formula for the nth term of the sequence 900, 450, 225.

The nth term is a_n = 900 × (0.5)^{n-1}.

What is the pattern followed in the sequence 900, 450, 225?

Each term is obtained by multiplying the previous term by 0.5, indicating a geometric sequence with a common ratio of 0.5.

Can the sequence 900, 450, 225 be classified as an arithmetic sequence?

No, because the differences between terms are not constant; it is a geometric sequence.

What is the 5th term in the sequence 900, 450, 225?

Using the formula a_n = 900 × (0.5)^{n-1}, the 5th term is 900 × (0.5)^{4} = 900 × 0.0625 = 56.25.

How do you determine whether a sequence is arithmetic or geometric?

By checking if the difference between consecutive terms is constant (arithmetic) or if the ratio of consecutive terms is constant (geometric).

Write a recursive rule for the sequence 900, 450, 225.

The recursive rule is a_1 = 900, and for n > 1, a_n = a_{n-1} × 0.5.

Classify The Sequence As Arithmetic Or Geometric; Then Write A Rule For The Nth Term. 900,450,225,

Classify The Sequence As Arithmetic Or Geometric; Then Write A Rule For The Nth Term. 900, 450, 225,

Introduction

Sequences are fundamental concepts in mathematics, representing ordered lists of numbers generated by specific rules. Recognizing whether a sequence is arithmetic or geometric is crucial for understanding its behavior and for deriving formulas to find any term within the sequence. In this article, we will analyze the sequence 900, 450, 225, and determine whether it is arithmetic or geometric. Subsequently, we will develop a general rule, known as the explicit formula, for finding the nth term of the sequence.

Understanding Sequences: Arithmetic and Geometric

What Is an Arithmetic Sequence?

An arithmetic sequence is a sequence of numbers where the difference between consecutive terms is constant.

This constant difference is called the common difference, denoted by \( d \).

The general form of an arithmetic sequence is:
\( an = a1 + (n - 1)d \)
where:

What Is a Geometric Sequence?

A geometric sequence is a sequence where each term is obtained by multiplying the previous term by a fixed number called the common ratio, denoted by \( r \).

The general form of a geometric sequence is:
\( an = a1 \times r^{n-1} \)
where:

Analyzing the Given Sequence: 900, 450, 225

Step 1: Check for Arithmetic Pattern

To determine if the sequence is arithmetic, we examine the differences between consecutive terms:

450 - 900 = -450

225 - 450 = -225

Since the differences are not constant (-450 and -225), the sequence is not arithmetic.

Step 2: Check for Geometric Pattern

To verify if the sequence is geometric, we calculate the ratio of successive terms:

450 ÷ 900 = 0.5

225 ÷ 450 = 0.5

Both ratios are equal to 0.5, indicating a consistent ratio. Therefore, the sequence is geometric with a common ratio \( r = 0.5 \).

Classifying the Sequence

Based on the analysis, the sequence 900, 450, 225 is a geometric sequence because the ratio between successive terms is constant at 0.5. It is not an arithmetic sequence because the differences between terms are not constant.

Writing the Nth Term Rule

Step 1: Identify the First Term and Common Ratio

First term, \( a_1 = 900 \)

Common ratio, \( r = 0.5 \)

Step 2: Use the Geometric Sequence Formula

The general formula for the nth term of a geometric sequence is:
\( an = a1 \times r^{n-1} \)

Substituting the known values:
\( a_n = 900 \times (0.5)^{n-1} \)

Explicit Formula for the Sequence

The explicit rule for the nth term of the sequence 900, 450, 225, ... is:
\( a_n = 900 \times (0.5)^{n-1} \)

Verifying the Formula

Calculate the first few terms:

When \( n=1 \):
\( a_1 = 900 \times (0.5)^{0} = 900 \times 1 = 900 \) ✓
When \( n=2 \):
\( a_2 = 900 \times (0.5)^{1} = 900 \times 0.5 = 450 \) ✓
When \( n=3 \):
\( a_3 = 900 \times (0.5)^{2} = 900 \times 0.25 = 225 \) ✓

The formula accurately generates the sequence's terms, confirming its correctness.

Additional Insights and Applications

Graphing the Sequence

Plotting the sequence's terms against their positions (n-values) would show a decreasing exponential curve approaching zero, characteristic of a geometric sequence with a ratio less than 1.

Real-World Applications

Radioactive decay, where substances decrease exponentially over time.

Financial calculations involving depreciation or compound interest with decreasing balances.

Population models where resources or populations decrease by a consistent ratio.

Summary

In conclusion, the sequence 900, 450, 225 is a geometric sequence with a common ratio of 0.5. Its nth term can be expressed explicitly by the formula:
\( a_n = 900 \times (0.5)^{n-1} \). This formula allows us to determine any term in the sequence efficiently, aiding in analysis and application across various fields.

Final Remarks

Recognizing whether a sequence is arithmetic or geometric is a foundational skill in mathematics. It involves examining the differences and ratios of successive terms. Once identified, formulating an explicit rule for the nth term becomes straightforward, enabling deeper understanding and practical computations involving the sequence.