Finding the 12th Term of a Geometric Series: A Comprehensive Guide

Geometric series, a fundamental concept in mathematics, involve a sequence where each term is obtained by multiplying the previous term by a constant factor, known as the common ratio. This article will guide you through the process of finding the 12th term of a geometric series given the sum of the first 12 terms and the sum of the first 11 terms.

Understanding Geometric Series

A geometric series is defined as 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. For example, if the first term is denoted by (a_1) and the common ratio by (r), the terms of the geometric series can be written as:

[a_1, a_1r, a_1r^2, a_1r^3, ldots]

Determining the 12th Term

The problem at hand involves finding the 12th term of a geometric series where the sum of the first 12 terms is 797160, and the sum of the first 11 terms is 265719. This can be solved by subtracting the sum of the first 11 terms from the sum of the first 12 terms.

Step-by-Step Solution

Identify the given information:

Sum of the first 12 terms, (S_{12} 797160) Sum of the first 11 terms, (S_{11} 265719)

Calculate the 12th term (a_{12}) by subtracting (S_{11}) from (S_{12}):

[a_{12} S_{12} - S_{11}]

Perform the subtraction:

[a_{12} 797160 - 265719 531441]

Identify the common ratio (r):

Given that the term (a_{12} 531441), we can determine the common ratio. For a geometric series, the terms follow the pattern (a_1, a_1r, a_1r^2, ldots).

Determine the common ratio (r):

To find (r), we can use the fact that the terms of the series are multiplied by (r) successively. Start by assuming the first term (a_1 3), and check if it fits the pattern:

Verify the first 11 terms:

[3, 9, 27, 81, 243, 729, 2187, 6561, 19683, 59049, 177147]

Calculate the sum of the first 11 terms:

[3 9 27 81 243 729 2187 6561 19683 59049 177147 265719]

Identify the 12th term:

Using the determined common ratio (r 3), the 12th term is:

Note the pattern:

[177147 times 3 531441]

Summarize the 12th term:

[a_{12} 531441]

Conclusion

In conclusion, the 12th term of the given geometric series, where the sum of the first 12 terms is 797160 and the sum of the first 11 terms is 265719, is 531441. This result can be obtained by subtracting the sum of the first 11 terms from the sum of the first 12 terms. The common ratio of the series is found to be 3.

Related Keywords

geometric series, 12th term, common ratio