Finding the 8th Term of a Geometric Progression

Understanding how to calculate the nth term of a geometric progression (GP) is a fundamental concept in mathematics. In this article, we will explore how to find the 8th term of a GP given the first term and common ratio.

Problem Statement

Given a GP with the first term (a_1 4) and a common ratio (r 2), we need to find the 8th term ((a_8)).

Solution

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

[ a_n a_1 r^{n-1} ]

Substituting the given values:

[ a_8 a_1 r^{8-1} 4 cdot 2^7 ]

Calculating the value:

[ 4 cdot 2^7 4 cdot 128 512 ]

Therefore, the 8th term of the GP is 512.

Explanation

Let's break down the solution step by step:

The first term (a_1 4). The common ratio (r 2). The formula for the nth term of a GP is (a_n a_1 r^{n-1}). To find the 8th term, we use the formula as follows: Substitute (n 8), (a_1 4), and (r 2): [ a_8 4 cdot 2^{8-1} 4 cdot 2^7 ] Calculate (2^7), which is 128. Multiply 4 by 128 to get 512.

Using the GP Formula

The general formula for the nth term of a GP is:

[ a_n a_1 r^{n-1} ]

In this case:

(a_1 4) (r 2) (n 8)

Substituting these values:

[ a_8 4 cdot 2^{8-1} 4 cdot 2^7 4 cdot 128 512 ]

Additional Insights

For any GP with the first term (a_1) and common ratio (r), the nth term can be determined using the same formula:

[ a_n a_1 r^{n-1} ]

This is a powerful tool to solve a variety of GMAT, SAT, and other mathematical problems involving geometric progressions.

Key Takeaways

Identify the first term and the common ratio. Use the formula (a_n a_1 r^{n-1}). Substitute the appropriate values for (a_1), (r), and (n). Multiply the terms to get the nth term.

Conclusion

Finding the 8th term of a GP is a straightforward process that involves understanding the relationship between terms in a geometric progression. By using the formula (a_n a_1 r^{n-1}), we can easily determine the 8th term in this case, which is 512.