Finding the 10th Term of a Geometric Progression

Geometric progressions (GPs) are sequences where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio. In this article, we will explore how to find the 10th term of a GP given the 5th and 8th terms.

Understanding the Sequence

Let's start with the given information: the 5th term of the GP is 4 1/2, and the 8th term is 15 3/16. These terms can be represented as improper fractions for easier manipulation.

Expressing as Improper Fractions

First, let's convert the mixed numbers into improper fractions:

4 1/2 9/2 15 3/16 243/16

Now, we need to find the common ratio r by analyzing the terms. The common ratio in a GP can be found by dividing any term by the term preceding it.

Calculating the Common Ratio

Given the 5th term is 9/2 and the 8th term is 243/16, we can calculate the common ratio as follows:

(a8) / (a5)  (243/16) / (9/2)

To simplify this, we first rewrite 9/2 as 72/16:

(243/16) / (72/16)  243/72

Next, we simplify 243/72:

243 / 72  243 / 24  27 / 8

Therefore, the common ratio r is 27/8. However, we notice that this can be simplified further to 3/2. This simplification is crucial as it indicates that the common ratio is consistent throughout the GP.

Calculating the 10th Term

We can now use this common ratio to find the 10th term. The 8th term we have is 243/16. To find the 10th term, we multiply the 8th term by the common ratio squared (since the 10th term is two more terms ahead of the 8th term).

Let's calculate it step-by-step:

Common ratio squared: (3/2) * (3/2) 9/4 10th term: (243/16) * (9/4) 2187/64

The final answer is the 10th term being 2187/64, which can be written as a mixed number when desired.

Note: The initial calculation implied an error in the process, as the common ratio 3/2 was used more directly and intuitively, rather than the more complex calculation initially suggested.

Conclusion

In this article, we have demonstrated how to find the 10th term of a geometric progression when given the 5th and 8th terms. Understanding the common ratio and using it to calculate subsequent terms is key to solving such problems.