Introduction

Understanding the properties of geometric sequences is crucial in various fields of mathematics and its applications. In this article, we explore a specific geometric sequence problem where the sum to infinity is given as 81, and the sum of all odd-numbered terms is 121.5. Through this, we will determine the common ratio and the first term of the sequence.

The Problem Statement

We are provided with the sum to infinity of a geometric sequence as 81 and the sum of all odd-numbered terms of the sequence as 121.5. We need to find the common ratio (r) and the first term (u1) of the sequence.

Mathematical Formulations

The full series sum is given by:

S frac{a}{1-r}S frac{a}{1-r}

Note that the odd-numbered terms form a GP with the same first term but a common ratio of ( r^2 ). The sum of the odd terms is given by:

S_{odd} frac{a}{1-r^2}S_{odd} frac{a}{1-r^2}

The even terms, on the other hand, form a GP with a common ratio of ( r^2 ) but with a first term of ( ar ). The sum of the even terms is therefore given by:

S_{even} frac{ar}{1-r^2}S_{even} frac{ar}{1-r^2}

Dividing the sum of the even terms by the sum of the odd terms, we get:

frac{S_{even}}{S_{odd}} rfrac{S_{even}}{S_{odd}} r

This can be rewritten as:

r frac{81-121.5}{121.5}r frac{81-121.5}{121.5}

Simplifying this expression, we find:

r frac{-40.5}{121.5}r frac{-40.5}{121.5}

-frac{81}{243} -frac{81}{243}

-frac{1}{3} -frac{1}{3}

Solving for the First Term

Now that we have the common ratio, we can substitute it back into the sum formula to find ( a ):

S frac{a}{1-r}S frac{a}{1-r}

Given ( S 81 ) and ( r -frac{1}{3} ):

81 frac{a}{1-(-frac{1}{3})}81 frac{a}{1-(-frac{1}{3})}

81 frac{a}{1 frac{1}{3}}81 frac{a}{1 frac{1}{3}}

81 frac{a}{frac{4}{3}}81 frac{a}{frac{4}{3}}

Therefore:

a 81 times frac{4}{3}a 81 times frac{4}{3}

a 81 times frac{4}{3}a 81 times frac{4}{3}

a 108a 108

Conclusion

The first term (u1) of the sequence is 108 and the common ratio is -(frac{1}{3}).

To summarize, by understanding the properties of the geometric sequence and applying the given sums, we were able to determine the first term and the common ratio. This problem not only tests the understanding of geometric sequence concepts but also emphasizes the importance of algebraic manipulation in problem-solving.