Understanding Geometric Progressions: Finding the 10th Term

Geometric progressions are a fundamental concept in mathematics, often encountered in various fields such as finance, physics, and computer science. A geometric progression, or geometric sequence, is a list 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 (r). This article will guide you through finding the 10th term of the given geometric sequence: 1, 3, 9, 27, using both mathematical concepts and programming techniques.

The Mathematical Approach

Given the first term a1 1 and the common ratio r 3, the general formula for the nth term of a geometric progression is:

Tn a1 * rn-1

Calculating the 10th Term Manually

To find the 10th term, we need to substitute N 10, a1 1, and r 3 into the formula:

T10 1 * 310-1 39

Calculating 39 manually:

31 3 32 9 33 27 34 81 35 243 36 729 37 2187 38 6561 39 19683

Thus, the 10th term T10 19683.

Interactive Demonstration Using J Programming Language

The J programming language provides a simple and elegant way to generate the terms of a geometric progression. The code `3^i.11` generates the first 11 terms of the sequence starting from 1:

3^i.11

This outputs:

1 3 9 27 81 243 729 2187 6561 19683 59049

We can see from this list that the 10th term is indeed 19683, and the 11th term is 59049. Let's re-verify the 10th term in detail:

10th term: 1 * 39 19683

Using the Geometric Progression Formula in Other Contexts

The geometric progression formula is not limited to manual calculations or programming. It can be used in various real-world applications, such as finance, population growth, and signal processing. For example, in finance, it can be used to calculate compound interest over time.

Here are a few additional calculations using the general term formula Tn a1 * rn-1 for different values of n:

5th term: T5 1 * 35-1 1 * 34 81 7th term: T7 1 * 37-1 1 * 36 729 11th term: T11 1 * 311-1 1 * 310 59049

Conclusion

Geometric progressions are powerful tools for modeling various phenomena in science and technology. By understanding the general formula and using programming languages like J, we can easily find any term in a geometric sequence. In this case, the 10th term of the sequence 1, 3, 9, 27 is 19683. Whether you are a student, a professional, or just curious about mathematical sequences, mastering these concepts will greatly enhance your problem-solving skills.