Geometric Series: Sum Of Even Terms Explained

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Let's dive into a fun math problem involving geometric series! If you're scratching your head over infinite sums and even-numbered terms, don't worry, I'm here to break it down for you. We're going to explore how to find the sum of all the even-numbered terms in a geometric series, given that the infinite sum is 4 and the first term is 43\frac{4}{3}. Buckle up, because math can actually be pretty cool when you understand what's going on!

Understanding Geometric Series

Before we jump into solving the problem, let's make sure we're all on the same page about what a geometric series actually is. A geometric series is essentially the sum of a sequence where each term is multiplied by a constant ratio to get the next term. Think of it like this: you start with a number, and then you keep multiplying it by the same thing over and over again. The sum of all those numbers is the geometric series.

The general form of a geometric series looks like this:

a+ar+ar2+ar3+...\qquad a + ar + ar^2 + ar^3 + ...

Where:

  • a is the first term.
  • r is the common ratio.

The sum to infinity of a geometric series (when |r| < 1) is given by the formula:

S=a1−r\qquad S = \frac{a}{1 - r}

This formula is super important because it tells us that if the common ratio r is between -1 and 1, the series will converge to a finite sum, even if it goes on forever. If r is greater than or equal to 1 or less than or equal to -1, the series diverges, meaning it doesn't have a finite sum.

So, to recap, a geometric series is a sequence of numbers where each term is found by multiplying the previous term by a constant ratio. The sum of this series to infinity can be calculated using a simple formula, provided that the absolute value of the common ratio is less than 1. Got it? Great, let's move on!

Setting Up the Problem

Now that we've got a handle on what geometric series are all about, let's get back to the specific problem we're trying to solve. We know two key pieces of information:

  1. The sum to infinity of the geometric series is 4.
  2. The first term, a, is 43\frac{4}{3}.

Our goal is to find the sum of all the even-numbered terms in this series. That means we want to find the sum of the second term, the fourth term, the sixth term, and so on. In other words, we're looking for:

ar+ar3+ar5+...\qquad ar + ar^3 + ar^5 + ...

To find this sum, we'll first need to figure out the common ratio, r. We can do this using the formula for the sum to infinity of a geometric series:

S=a1−r\qquad S = \frac{a}{1 - r}

We know that S = 4 and a = $\frac{4}{3}$, so we can plug those values into the formula and solve for r:

4=431−r\qquad 4 = \frac{\frac{4}{3}}{1 - r}

Let's solve this equation step by step:

4(1−r)=43\qquad 4(1 - r) = \frac{4}{3}

1−r=13\qquad 1 - r = \frac{1}{3}

r=1−13\qquad r = 1 - \frac{1}{3}

r=23\qquad r = \frac{2}{3}

So, the common ratio r is 23\frac{2}{3}. Now that we know a and r, we're one step closer to finding the sum of the even-numbered terms!

Finding the Sum of Even-Numbered Terms

Alright, we've successfully found the common ratio, r, which is 23\frac{2}{3}. Now, let's zoom in on those even-numbered terms. Remember, we're trying to find the sum of the series:

ar+ar3+ar5+...\qquad ar + ar^3 + ar^5 + ...

Notice anything interesting about this series? Well, it's actually another geometric series! The first term is ar, and the common ratio is r^2. Let's break that down:

  • First term: The first even-numbered term is ar. Since a = $\frac{4}{3}$ and r = $\frac{2}{3}$, the first term is 43â‹…23=89\frac{4}{3} \cdot \frac{2}{3} = \frac{8}{9}`.
  • Common Ratio: To get from one even-numbered term to the next, we're multiplying by r^2. So, the common ratio for this new series is (23)2=49(\frac{2}{3})^2 = \frac{4}{9}`.

Now that we know the first term and the common ratio of the series of even-numbered terms, we can use the sum to infinity formula again!

Seven=ar1−r2\qquad S_{even} = \frac{ar}{1 - r^2}

Plugging in the values we found:

Seven=891−49\qquad S_{even} = \frac{\frac{8}{9}}{1 - \frac{4}{9}}

Seven=8959\qquad S_{even} = \frac{\frac{8}{9}}{\frac{5}{9}}

Seven=89â‹…95\qquad S_{even} = \frac{8}{9} \cdot \frac{9}{5}

Seven=85\qquad S_{even} = \frac{8}{5}

So, the sum of all the even-numbered terms in the geometric series is 85\frac{8}{5} or 1.6. Awesome! We did it!

Conclusion

So, there you have it! We've successfully navigated the world of geometric series to find the sum of all the even-numbered terms in a series where the infinite sum is 4 and the first term is 43\frac{4}{3}. Remember, the key steps were:

  1. Understanding Geometric Series: Grasping the concept of a geometric series and its sum to infinity formula.
  2. Finding the Common Ratio: Using the given information to calculate the common ratio, r.
  3. Identifying the New Series: Recognizing that the even-numbered terms form another geometric series with a first term of ar and a common ratio of r^2.
  4. Applying the Sum to Infinity Formula: Using the formula to find the sum of the new series.

By breaking down the problem into smaller, manageable steps, we were able to solve it without too much trouble. Keep practicing, and you'll become a pro at solving geometric series problems in no time! Math isn't so scary after all, is it?