Geometric Progression: Finding the 4th Term of a Sequence

Geometric progression (GP) is a fundamental concept in mathematics, widely used in various fields such as physics, engineering, and finance. This article aims to explain the process of finding the 4th term of a GP, focusing on the specific example where the first term a1 is 2 and the common ratio r is 3.

Understanding the Geometric Progression (GP)

A geometric progression is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. Mathematically, this can be represented as:

an a1 rn-1

Where:

an is the nth term of the progression, a1 is the first term, r is the common ratio, n is the term number.

Finding the 4th Term (A4) in the GP

Given:

The first term a1 is 2. The common ratio r is 3.

To find the 4th term (A4) of the GP, we use the formula:

A4 a1 r4-1

Substituting the given values:

A4 2 × 33

Calculating the value:

A4 2 × 27 54

Therefore, the 4th term A4 in the given geometric progression is 54.

Visualizing the Sequence

The sequence generated from this GP can be visualized as:

2 2 × 3 6 6 × 3 18 18 × 3 54 54 × 3 162 162 × 3 486

Thus, the sequence is 2, 6, 18, 54, 162, 486, ...

Conclusion

This article has provided a detailed step-by-step guide on finding the 4th term of a geometric progression with a first term of 2 and a common ratio of 3. Understanding this concept is crucial for anyone dealing with sequences and series in mathematics and related fields.

Further Reading

For more detailed information on geometric progressions and related topics, refer to the following resources:

Geometric Progressions on MathIsFun Geometric Progression on BYJUS