Determining the 11th Term of a Geometric Sequence

Understanding and manipulating geometric sequences is a fundamental concept in mathematics, especially in fields such as finance, physics, and computer science. In this article, we'll explore how to find the 11th term of a geometric sequence, given the 3rd and 7th terms. We'll provide a step-by-step solution and a detailed explanation with relevant keywords.

Introduction to Geometric Sequences

A geometric sequence is a sequence 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, denoted as (r).

Problem Statement

Given the 3rd term (a_3 20) and the 7th term (a_7 320) of a geometric sequence, we want to find the 11th term (a_{11}).

Solution

Step 1: Identify the Formula for the nth Term

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

(a_n a_1 r^{n-1})

Where:

(a_n) is the nth term, (a_1) is the first term, (r) is the common ratio, (n) is the term number.

Step 2: Set Up Equations for Given Terms

Given:

(a_3 a_1 r^2 20) (a_7 a_1 r^6 320)

Step 3: Solve for the Common Ratio (r)

Taking the ratio of the 7th term to the 3rd term:

(frac{a_7}{a_3} frac{a_1 r^6}{a_1 r^2} r^4 frac{320}{20} 16)

(r r^4 2)

Step 4: Solve for the First Term (a_1)

Using the value of (r 2) in the equation for the 3rd term:

(a_1 r^2 20)

(a_1 (2)^2 20)

(a_1 frac{20}{4} 5)

Step 5: Calculate the 11th Term (a_{11})

Using the formula for the nth term:

(a_{11} a_1 r^{10} 5 r^{10})

(a_{11} 5 (2)^{10} 5 times 1024 5120)

Conclusion

The 11th term of the geometric sequence is (5120). This solution demonstrates the importance of using the properties of geometric sequences and the step-by-step process to solve for unknown terms.

Additional Information and Key Concepts

Geometric Sequence: A sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio.
Common Ratio: The fixed number that each term in a geometric sequence is multiplied by the previous term to get the next term.
Nth Term: The term at position (n) in a geometric sequence.