Geometric Sequences: Exploring Patterns and Calculations

Geometric sequences are sequences where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. This article will explore how to find specific terms within a geometric sequence, using the given problem as an example. We will calculate the 8th term of a geometric sequence where the third term is 25 and the common ratio is 4. Additionally, we will provide a geometric progression (GP) for 10 terms and discuss how to find the total sum of the sequence.

Understanding the Problem

Let's take the given problem: What is the 8th term of a geometric sequence where the third term is 25 and the common ratio is 4? To solve this, we need to use the formula for the nth term of a geometric sequence:

Formula and Calculation

The formula for the nth term of a geometric sequence is given by:

a_n a_1 cdot r^{n-1}

However, we are given the third term, so we can use:

a_n a_3 cdot r^{n-3}

Step 1: Finding the 8th Term

We know that:

a_8 a_3 cdot r^{8-3} 25 cdot 4^5

Calculating this step by step:

4^5 1024 25 cdot 1024 25600

So, the 8th term of the sequence is:

a_8 25600

Geometric Sequence Example

Now, let's look at a complete geometric progression with 10 terms, where the third term is 25 and the common ratio is 4:

Term Number Term Value 1 1.5625 2 6.25 3 25 4 100 5 400 6 1600 7 6400 8 25600 9 102400 10 409600

As you can see, the 8th term is 25600, which we calculated earlier.

Sum of the Sequence

The sum of the first n terms of a geometric sequence can be calculated using the formula:

S_n a_1 cdot frac{1 - r^n}{1 - r}

For our sequence, where the first term (a_1) is 1.5625, the common ratio (r) is 4, and we want the sum of the first 10 terms:

S_10 1.5625 cdot frac{1 - 4^{10}}{1 - 4}

Calculating this:

4^{10} 1048576 1 - 1048576 -1048575 1 - 4 -3 S_10 1.5625 cdot frac{-1048575}{-3} 1.5625 cdot 349525 546132.8125

The total sum of the sequence is:

546132.8125

Conclusion

This example demonstrates the process of finding specific terms and the total sum in a geometric sequence. By understanding the formula and the steps involved, you can solve similar problems and apply this knowledge to various mathematical and real-world scenarios.