Introduction

A geometric sequence is a series of numbers in which each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. The nth term of a geometric sequence can be found using a simple formula, making it a valuable tool in various fields such as mathematics, finance, and technology.

The General Formula for the nth Term

The nth term of a geometric sequence can be given by the formula:

an a rn-1

Where:

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

This formula is derived by observing the pattern in the sequence. Each term is a multiple of the previous term by the common ratio.

Example: Calculating the 5th Term

Consider the geometric sequence: 3, 6, 12, 24, ...

First term (a1): 3 Common ratio (r): 2 We want to find the 5th term (n 5):

To find the 5th term, use the formula:

a5 3 2(5-1)

Calculating the exponent first:

24 16

Then:

a5 3 16 48

Thus, the 5th term is 48.

Using Two Given Terms to Find the nth Term

If you are given two terms of the geometric sequence, you can use them to find the nth term. For example, if you know the 4th term and the 7th term:

a4 10 a7 200

The difference in indices is 3, so:

a7 a4 r3

Substitute the known values:

200 10 r3

Solving for r3:

r3 20

Taking the cube root on both sides:

r 201/3

Once you have the common ratio, you can use it to find the first term. Since:

a4 a1 r3

Substituting the known values:

10 a1 20

Solving for a1:

a1 10 / 20 1 / 2

Now you have the first term and the common ratio, so you can find the nth term using the formula an a1 rn-1.

Conclusion

Understanding how to find the nth term of a geometric sequence is a crucial skill in many areas of mathematics and practical applications. Whether you are dealing with financial growth, population dynamics, or technological advancements, the formula and steps outlined above can help you determine any term in the sequence accurately and efficiently.