Unraveling A Complex Math Problem: Detailed Solution!

Hey guys! Let's dive into a pretty interesting math problem today. We're gonna calculate the value of a rather long expression, and I'll break it down step-by-step so it's super clear. The problem looks a bit intimidating at first glance, with fractions and series, but trust me, we can totally handle it! So, what's the deal? We need to find the value of 'a', which is defined by this equation:

a = 8/√7 + 9/14 + 10/21 + 11/28 + ... + 259/1764 - (1 + 1/2 + 1/3 + 1/4 + ... + 1/252)

See, not so scary, right? Let's start dissecting this beast! The key is to recognize patterns and use some clever tricks to simplify things. Our journey involves a combination of fraction manipulation, series understanding, and a touch of patience. Ready to roll up your sleeves? Let's go!

Breaking Down the Series: A Closer Look

Alright, the first thing we want to do is take a closer look at the first part of the equation. That long series of fractions, from 8/√7 all the way to 259/1764. We need to figure out a way to simplify that before we get bogged down. Notice how the numerators (the top numbers) are increasing: 8, 9, 10, 11, and so on. The denominators (the bottom numbers) are also increasing, but in a slightly more complicated manner: 7, 14, 21, 28, and so on. These are all multiples of 7, of course, but that is where our trick comes in.

Notice that the denominator of each fraction is 7 times by n, where n is a sequence of natural numbers that starts from 1. 8/√7 -> 7 * 1 = 7 9/14 -> 7 * 2 = 14 10/21 -> 7 * 3 = 21 11/28 -> 7 * 4 = 28

And the numerator is 7 + n, for the same n.

8/√7 -> 7 + 1 = 8 9/14 -> 7 + 2 = 9 10/21 -> 7 + 3 = 10 11/28 -> 7 + 4 = 11

So, in a general form of each term, we can rewrite each term in the first part of the equation in a more general form. This will help us to deal with the whole series without writing each fraction individually.

Each term in the series 8/7 + 9/14 + 10/21 + 11/28 + ... + 259/1764 can be written as (7 + n) / (7n), where n varies from 1 to 252. Where n is the term's index.

This is where the magic happens! We can rewrite each fraction in the series like this:

(7 + n) / (7n) = 7/(7n) + n/(7n) = 1/n + 1/7

So, our series becomes:

8/7 + 9/14 + 10/21 + 11/28 + ... + 259/1764 = (1/1 + 1/7) + (1/2 + 1/7) + (1/3 + 1/7) + ... + (1/252 + 1/7)

See how we're simplifying already? The main series, which looked so complicated, has now been transformed into something that's much more manageable. That's the power of recognizing the pattern and applying some algebra.

Simplifying the Expression Further: Grouping Terms

Now, let's take this simplified series and group the terms a bit differently. We'll separate the 1/n terms from the 1/7 terms. Doing this will help us to easily compare it with the second part of the equation.

(1/1 + 1/7) + (1/2 + 1/7) + (1/3 + 1/7) + ... + (1/252 + 1/7)

This can be rewritten as:

(1/1 + 1/2 + 1/3 + ... + 1/252) + (1/7 + 1/7 + 1/7 + ... + 1/7)

Notice anything cool? The first part of this rewritten series is the same as the second series in the original problem, 1 + 1/2 + 1/3 + 1/4 + ... + 1/252. Let's move on to the next step.

Tackling the Second Series: The Harmonic Series

Now let's get to the second part of the equation: 1 + 1/2 + 1/3 + 1/4 + ... + 1/252. This type of series is called a harmonic series. While we don't need to calculate the exact sum of this series (thankfully!), recognizing it is helpful. However, if we need to find its sum, we can approximate it by this formula:

H(n) = ln(n) + γ

Where γ is the Euler–Mascheroni constant (approximately 0.57721).

In our problem, we don't need to use the formula. We can use the simplification from the previous step. Let's write the original equation again, using the transformations we did before.

a = (8/7 + 9/14 + 10/21 + 11/28 + ... + 259/1764) - (1 + 1/2 + 1/3 + 1/4 + ... + 1/252)

And substitute with our rewritten series:

a = [(1/1 + 1/2 + 1/3 + ... + 1/252) + (1/7 + 1/7 + 1/7 + ... + 1/7)] - (1 + 1/2 + 1/3 + 1/4 + ... + 1/252)

Oh, look! Now we can see that the first part in the parentheses, the harmonic series, cancels out with the second part. This is wonderful! It makes the problem much easier.

Calculating the Remaining Sum

So, what remains? We just need to calculate (1/7 + 1/7 + 1/7 + ... + 1/7). How many terms are there in the series? Remember that each term of the series 8/7 + 9/14 + 10/21 + 11/28 + ... + 259/1764 can be written as (7 + n) / (7n), where n varies from 1 to 252. Thus, the series has 252 terms. So, we have 252 terms of 1/7.

a = (252 * 1/7)

Simple math: 252/7 = 36.

The Final Answer: Voila!

So, our answer is: a = 36. Isn't it amazing how a complicated-looking problem can be broken down into simpler parts? We used a bit of algebraic manipulation, pattern recognition, and the power of series to find our solution. It's like solving a puzzle – you just have to find the right pieces and fit them together. Great job, guys! I hope this explanation was helpful. Let me know if you have any questions.

In short:

• Recognize the pattern: Identify the relationship between the numerators and denominators.
• Simplify the series: Rewrite the main series to make it manageable.
• Group terms: This helps to make cancellations.
• Calculate: Find the final sum.

And there you have it! A complete solution to this math problem. Keep practicing and you'll become a pro in no time!