Identifying the First Term of a Geometric Sequence: 9, 27, 81

Understanding geometric sequences is a fundamental concept in mathematics. This article will guide you through the process of identifying the first term of a specific geometric sequence: 9, 27, 81, using various methods and explanations.

Introduction to Geometric Sequences

A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio (r). The formula for the nth term of a geometric sequence is:

a_n a_1 r^{n-1}

Where:

a_n is the nth term of the sequence, a_1 is the first term, r is the common ratio, and n is the term number.

Identifying the Common Ratio

To find the common ratio, we can divide any term by its preceding term:

(81/27 3) (27/9 3) (9/3 3)

Each division results in the same value, which confirms that the common ratio, (r 3).

Calculating the First Term

Given the formula for the nth term of a geometric sequence, we can use it to find the first term, (a_1). By setting (n 1) in the formula:

(a_1 a_n / r^{n-1} 9 / 3^{1-1} 9 / 3^0 9 / 1 9)

Verification and Generalization

Let's verify the first term, 3, using the formula:

(a_n a_1 r^{n-1})

For the 3rd term ((n 3)):

(a_3 3 * 3^{3-1} 3 * 3^2 3 * 9 27)

This matches the given sequence, confirming our calculations.

The GP Sequence

The geometric sequence can be written as:

3, 9, 27, 81, …

Each term can be generated by multiplying the previous term by the common ratio, 3:

3 * 3 9 9 * 3 27 27 * 3 81

Conclusion

The first term of the geometric sequence 9, 27, 81 is 3. This demonstrates how we can use the common ratio and the formula for the nth term to identify the first term. Understanding geometric sequences is crucial for various applications in mathematics and related fields.

Keywords

This article focuses on the following key terms:

Geometric Sequence First Term Common Ratio