Calculating Series Sums: A Step-by-Step Guide

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Hey guys! Let's dive into a fun math problem today. We're going to figure out the sum of an interesting series. It's a great opportunity to brush up on our algebra skills and see how different parts of a series can come together. We'll break down the problem step by step to make sure everyone understands the process. This specific type of problem, involving series summation, is super common in algebra, so understanding how to approach it is key. Knowing how to deal with infinite or repeating series can really help with more advanced mathematical concepts down the line. We will be using the concepts of geometric progressions and their sums to solve the given problem. Let's get started!

Understanding the Problem: The Series Unveiled

First things first, let's take a good look at the series we're dealing with. The series is:

38+1+964+14+27512+116+...\frac{3}{8}+1+\frac{9}{64}+\frac{1}{4}+\frac{27}{512}+\frac{1}{16}+...

Our mission is to find the sum of this series. At first glance, it might seem a bit intimidating, but trust me, we can break it down into manageable chunks. The series includes fractions, which might give you flashbacks to elementary school, but don't worry, we're going to use some clever tricks to simplify things. The main goal is to identify a pattern or a structure that allows us to calculate the sum efficiently. If you are good with patterns and identifying the type of progression, then this should be fun to solve. This particular series is a mixture of terms, some of which appear to be following a specific pattern. Let us rearrange and group the terms and find the individual sums.

Before we start calculating, let's examine the options given to us. We have the following options:

  1. 114151\frac{14}{15}
  2. 112171\frac{12}{17}
  3. 214152\frac{14}{15}
  4. 115161\frac{15}{16}
  5. 111151\frac{11}{15}

We will examine all of these options to see which one fits our answer. Now, let's figure out how to approach the calculation and find the correct option. We have to identify the hidden patterns in the series to calculate its sum easily. Then, we will find which of the given options corresponds to our answer.

Deconstructing the Series: Identifying the Patterns

Alright, let's get our detective hats on and try to find some patterns. If we closely look at the series, we can see that it's made up of two different types of terms. One set of terms is a geometric progression, while the other set also appears to be a geometric progression, but simpler. By separating the series and calculating the sum of each, we can later combine them to get the final answer. Spotting these patterns is crucial for solving these kinds of problems. Let's see how we can separate the two geometric progressions.

Let's separate the given series into two separate series.

Series 1: 38+964+27512+...\frac{3}{8}+\frac{9}{64}+\frac{27}{512}+...

Here, the common ratio (r) can be found by dividing any term by its preceding term.

r=9/643/8=38r = \frac{9/64}{3/8} = \frac{3}{8}

Now, for an infinite geometric progression, the sum is given as:

S=a1βˆ’rS = \frac{a}{1-r}, where aa is the first term, and rr is the common ratio. So, for the first series,

S1=3/81βˆ’3/8=3/85/8=35S_1 = \frac{3/8}{1-3/8} = \frac{3/8}{5/8} = \frac{3}{5}

Series 2: 1+14+116+...1+\frac{1}{4}+\frac{1}{16}+...

Here also, we can find the common ratio (r) as before.

r=1/161/4=14r = \frac{1/16}{1/4} = \frac{1}{4}

So, for the second series,

S2=11βˆ’1/4=13/4=43S_2 = \frac{1}{1-1/4} = \frac{1}{3/4} = \frac{4}{3}

So, we have identified the pattern in the given series and separated it into two series. Now, let's calculate the sum of each and later combine them. After identifying the different geometric progressions, we will combine their sums to find the total sum. Now that we've identified the two series, calculating their sums will be easier. Remember, the key is to recognize the patterns and apply the appropriate formulas.

Summing Up: Combining the Results

Now that we've found the sum of both the individual series, it's time to put it all together. To find the total sum of the original series, we simply need to add the sums of the two series we identified earlier.

Total Sum = S1+S2S_1 + S_2

Total Sum = 35+43=9+2015=2915\frac{3}{5} + \frac{4}{3} = \frac{9+20}{15} = \frac{29}{15}

Let's convert this improper fraction into a mixed fraction for easier comparison with the options given.

2915=11415\frac{29}{15} = 1\frac{14}{15}

So, the total sum of the series is 114151\frac{14}{15}. We've successfully calculated the sum of the series by identifying the patterns, calculating individual sums, and then combining the results. This is a common strategy when dealing with complex series. Remember, understanding the underlying patterns is the key to solving these types of problems. This systematic approach ensures that we don't miss any terms and accurately determine the total sum. This is a good example of how understanding mathematical concepts can make complicated-looking problems quite manageable. We can now compare this value with the options given to us and mark our answer.

Choosing the Right Answer: Matching with the Options

Finally, we need to match our calculated sum with the given options to find the correct answer. We calculated the sum to be 114151\frac{14}{15}. Let's see which option matches our calculated value.

The options provided are:

  1. 114151\frac{14}{15}
  2. 112171\frac{12}{17}
  3. 214152\frac{14}{15}
  4. 115161\frac{15}{16}
  5. 111151\frac{11}{15}

By comparing our calculated sum with the given options, it's clear that the correct option is the one that matches our result, which is:

  1. 114151\frac{14}{15}

So, the correct answer is option 1. We did it, guys! We successfully found the sum of a complex series by breaking it down into manageable parts. Keep practicing these types of problems to become more comfortable with series and their summation. Remember, practice makes perfect! We are now confident in our answer after identifying the type of progression and calculating their sums.

Final Thoughts

That's it, guys! We've successfully navigated through the problem, identified the patterns, calculated the sums, and found the correct answer. This entire process demonstrates the power of breaking down a complex problem into simpler steps. We have learned how to deconstruct and solve the series problem. I hope this detailed explanation helped you understand the process. Feel free to try more problems like this to build your confidence and skills. Keep exploring the exciting world of mathematics, and happy solving! We have successfully calculated the value, and the correct option is 114151\frac{14}{15}. The main thing to remember is to identify the pattern and apply the appropriate formula for solving the problem.

  • Key Takeaways:
    • Series summation involves identifying patterns.
    • Geometric progressions have a specific formula for their sum.
    • Breaking down the problem into smaller parts makes it easier to solve.
    • Comparing the answer with the given options is crucial.

Great job everyone! Keep up the amazing work!