https://file.aiquickdraw.com/mathbot/file/1726416719075o8po9zt3.jpg

The problem you provided is asking for the value of the following expression:

\[
\frac{1}{1 + \sqrt{2}} + \frac{1}{\sqrt{2} + \sqrt{3}} + \frac{1}{\sqrt{3} + \sqrt{4}} + \cdots + \frac{1}{\sqrt{63} + \sqrt{64}}
\]

This is a telescoping series, where many terms cancel out after simplifying the individual fractions. Let's break it down.

### Step-by-Step Simplification:

Each term in the series has the form:

\[
\frac{1}{\sqrt{n} + \sqrt{n+1}}
\]

To simplify this, multiply both the numerator and denominator by \(\sqrt{n+1} - \sqrt{n}\) to rationalize the denominator:

\[
\frac{1}{\sqrt{n} + \sqrt{n+1}} \times \frac{\sqrt{n+1} - \sqrt{n}}{\sqrt{n+1} - \sqrt{n}} = \frac{\sqrt{n+1} - \sqrt{n}}{(\sqrt{n+1})^2 - (\sqrt{n})^2} = \frac{\sqrt{n+1} - \sqrt{n}}{(n+1) - n} = \sqrt{n+1} - \sqrt{n}
\]

So, each term simplifies to \(\sqrt{n+1} - \sqrt{n}\).

### Applying this to the Series:

Now, the series becomes:

\[
(\sqrt{3} - \sqrt{2}) + (\sqrt{4} - \sqrt{3}) + (\sqrt{5} - \sqrt{4}) + \cdots + (\sqrt{64} - \sqrt{63})
\]

This is a telescoping series, meaning most terms cancel out. After cancellation, we are left with:

\[
-\sqrt{2} + \sqrt{64} = 8 - \sqrt{2}
\]

So, the value of the entire expression is \(8 - \sqrt{2}\).

### Approximating the Final Value:

We know \(\sqrt{2} \approx 1.414\), so:

\[
8 - \sqrt{2} \approx 8 - 1.414 = 6.586
\]

Rounding to the nearest whole number, the answer is approximately **7**, which corresponds to option (D).

---

Do you want further details or have any questions?

Here are 5 related questions you might find interesting:
1. How do you solve other types of telescoping series?
2. What are some methods for simplifying radical expressions?
3. Can you explain how rationalizing a denominator works for different cases?
4. How does the approximation of square roots affect results in math problems?
5. What is the significance of telescoping series in calculus?

**Tip:** When working with series, always check if it's telescoping to simplify it quickly!

What is the value of the expression: 1 / (1 + √2) + 1 / (√2 + √3) + 1 / (√3 + √4) + ... + 1 / (√63 + √64)?

Learn how to solve a telescoping series involving square roots, simplifying radical expressions and rationalizing denominators. This problem focuses on simplifying each term and using the telescoping nature of the series to find the final value. Suitable for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation