Infinite Geometric Series: Finding The Ratio Explained
Hey guys! Today, we're diving into a super interesting math problem involving infinite geometric series. If you've ever wondered how a series with an infinite number of terms can actually have a finite sum, you're in the right place! We're going to break down a specific problem step by step, so you'll not only understand the solution but also grasp the underlying concepts. Let's get started!
Understanding the Problem
Let's kick things off by stating the problem clearly. We're dealing with an infinite geometric series. This means we have a sequence of numbers where each term is found by multiplying the previous term by a constant value, known as the common ratio. The problem tells us two crucial things: the sum of the series is 405, and the second term in the series is 90. Our mission, should we choose to accept it, is to find the common ratio of this series. To understand this better, we need to know the formula for the sum of an infinite geometric series.
Key Concepts: Geometric Series and the Common Ratio
Before we dive into solving the problem, let's refresh our understanding of a few key concepts. A geometric series is a sequence where each term is multiplied by a constant value to get the next term. This constant value is called the common ratio, often denoted by 'r'. For example, the sequence 2, 6, 18, 54... is a geometric series because each term is obtained by multiplying the previous term by 3 (the common ratio). The general form of a geometric series is: a, ar, ar², ar³, ... where 'a' is the first term. Now, what happens when the series goes on infinitely? That's where things get interesting!
The Sum to Infinity Formula
Here’s the magic formula that makes solving our problem possible: The sum to infinity (S∞) of a geometric series is given by:
S∞ = a / (1 - r)
where:
- S∞ is the sum to infinity,
- a is the first term of the series, and
- r is the common ratio.
This formula holds true only when the absolute value of the common ratio (|r|) is less than 1. Why? Because if |r| ≥ 1, the terms in the series either stay the same size or get bigger, and the sum will keep growing without bound – it won't converge to a finite value. Think about it: if you keep adding larger and larger numbers, you'll never reach a limit!
Breaking Down the Given Information
Now that we've armed ourselves with the formula, let's revisit the information given in the problem. We know:
- The sum to infinity (S∞) is 405.
- The second term of the series is 90.
This is like having pieces of a puzzle. We need to figure out how these pieces fit together to reveal the common ratio. We're going to use the information about the second term to express it in terms of 'a' and 'r'. Remember, in a geometric series, the second term is simply the first term ('a') multiplied by the common ratio ('r'). So, we can write the second term as ar = 90. This equation is a crucial stepping stone in solving our problem.
Solving for the Ratio
Alright, let's roll up our sleeves and get into the nitty-gritty of solving for the common ratio. We have two key pieces of information: the sum to infinity formula and the value of the second term. We're going to use these two facts to create a system of equations and solve for our unknowns – the first term ('a') and, more importantly, the common ratio ('r').
Setting Up the Equations
From the sum to infinity formula, we have:
405 = a / (1 - r) (Equation 1)
And from the second term, we have:
ar = 90 (Equation 2)
We now have two equations with two unknowns. This is fantastic because it means we can solve for both 'a' and 'r'. There are a couple of ways we could approach this, but let's use a method that's both clear and efficient. We'll solve Equation 2 for 'a' and then substitute that expression into Equation 1. This will give us an equation with only 'r' as the unknown, which we can then solve directly.
Isolating 'a'
From Equation 2 (ar = 90), we can easily isolate 'a' by dividing both sides by 'r':
a = 90 / r (Equation 3)
This gives us an expression for 'a' in terms of 'r'. Now, we're going to take this expression and plug it into Equation 1. This process is called substitution, and it's a powerful technique for solving systems of equations.
Substituting and Simplifying
Now, substitute Equation 3 into Equation 1:
405 = (90 / r) / (1 - r)
This looks a bit messy, but don't worry, we'll simplify it step by step. The first thing to notice is that we have a fraction within a fraction. To get rid of this, we can multiply both the numerator and the denominator of the main fraction by 'r':
405 = 90 / (r * (1 - r))
Next, let's multiply both sides of the equation by r(1 - r) to get rid of the denominator:
405 * r * (1 - r) = 90
Now, divide both sides by 405:
r * (1 - r) = 90 / 405
Simplify the fraction on the right side:
r * (1 - r) = 2 / 9
Forming a Quadratic Equation
Let's expand the left side and rearrange the equation to get a standard quadratic form:
r - r² = 2 / 9
Multiply both sides by 9 to get rid of the fraction:
9r - 9r² = 2
Rearrange the terms to get the quadratic equation in the standard form (ax² + bx + c = 0):
9r² - 9r + 2 = 0
We now have a quadratic equation that we can solve for 'r'. There are a couple of ways to solve quadratic equations: factoring, using the quadratic formula, or completing the square. In this case, factoring is the easiest approach.
Solving the Quadratic Equation by Factoring
We need to find two numbers that multiply to (9 * 2 = 18) and add up to -9. These numbers are -6 and -3. So, we can factor the quadratic equation as follows:
(3r - 2)(3r - 1) = 0
Now, set each factor equal to zero and solve for 'r':
3r - 2 = 0 or 3r - 1 = 0
Solving these equations, we get:
r = 2/3 or r = 1/3
Choosing the Correct Ratio
We have two possible values for the common ratio: 2/3 and 1/3. However, we need to check if both of these values are valid in the context of our problem. Remember, for the sum to infinity formula to work, the absolute value of the common ratio must be less than 1 (|r| < 1). Both 2/3 and 1/3 satisfy this condition, so we need to consider both. But, let's consider the first term.
If r = 2/3, then using ar = 90, we get a * (2/3) = 90, so a = 135. This is a valid solution.
If r = 1/3, then using ar = 90, we get a * (1/3) = 90, so a = 270. This is also a valid solution.
Both values of 'r' fit the criteria. However, let's think about the implications. A common ratio of 2/3 means each term is multiplied by a larger fraction than if the common ratio were 1/3. This suggests that the series with r = 2/3 will converge to the sum of 405 faster than the series with r = 1/3. In most cases, problems of this type have a single unique solution that best fits all the conditions. So let's revisit our process to ensure we haven't missed anything. We’ve checked our math and the logic of the problem, and it seems we have two valid solutions. However, infinite geometric series problems often have subtle constraints. Since the question implies a single common ratio, let's look at what else the problem gives us and see if one of the ratios fits better practically. The series is infinite, which generally means the ratio will reduce the impact of each further term. The second term is given as 90. If r = 1/3, this means the series is decreasing faster than if r = 2/3. Because the sum of 405 is relatively close to 90, it suggests the terms don’t decrease too rapidly. Therefore, 2/3 is the more likely answer, as the terms decrease less sharply, contributing more to the sum.
Therefore, the common ratio of the geometric series is most likely 2/3.
Conclusion
So, there you have it! By understanding the concept of infinite geometric series, the sum to infinity formula, and applying some algebraic techniques, we successfully found the common ratio of the series. Remember, the key is to break down the problem into smaller, manageable steps, set up the equations correctly, and carefully solve for the unknowns. Math problems like these might seem daunting at first, but with a little practice and the right approach, you can conquer them like a pro. Keep practicing, keep exploring, and most importantly, keep enjoying the beauty of mathematics!