Consider the sequence {a_n} = 2 - \frac{1}{{{n^2} + \sqrt {{n^4} + \frac{1}{4}} }},  n ≥ 1
Evaluate: \frac{1}{2}\left[ {\sqrt {{a_1}}  + \sqrt {{a_2}}  + ..... + \sqrt {{a_{119}}} } \right]


To solve the given problem, we start by analyzing the sequence \( \{a_n\} \) defined by:

\[
a_n = 2 - \frac{1}{n^2 + \sqrt{n^4 + \frac{1}{4}}}
\]

### Step 1: Simplify the expression for \( a_n \)

First, observe the term inside the sequence:

\[
b_n = \frac{1}{n^2 + \sqrt{n^4 + \frac{1}{4}}}
\]

Let's simplify \( b_n \). The denominator can be rewritten to simplify using rationalization:

\[
n^2 + \sqrt{n^4 + \frac{1}{4}} = \frac{(n^2 + \sqrt{n^4 + \frac{1}{4}})(n^2 - \sqrt{n^4 + \frac{1}{4}})}{n^2 - \sqrt{n^4 + \frac{1}{4}}}
\]

Using the identity \( (a+b)(a-b) = a^2 - b^2 \), we find:

\[
= \frac{n^4 - (n^4 + \frac{1}{4})}{n^2 - \sqrt{n^4 + \frac{1}{4}}}
= \frac{-\frac{1}{4}}{n^2 - \sqrt{n^4 + \frac{1}{4}}}
\]

Thus, the denominator is effectively:

\[
n^2 + \sqrt{n^4 + \frac{1}{4}} = \frac{n^4 + \frac{1}{4}}{n^2 - \sqrt{n^4 + \frac{1}{4}}}
\approx 2n^2 \quad \text{(using approximation)}
\]

\[
b_n \approx \frac{1}{2n^2}
\]

### Step 2: Approximating \( a_n \)

Using the above approximation for \( b_n \):

\[
a_n \approx 2 - b_n \approx 2 - \frac{1}{2n^2}
\]

### Step 3: Finding the sum \( \frac{1}{2} \left( \sqrt{a_1} + \sqrt{a_2} + \ldots + \sqrt{a_{119}} \right) \)

Given the expression:

\[
\sqrt{a_n} \approx \sqrt{2 - \frac{1}{2n^2}}
\]

For large \( n \), \( \sqrt{2 - \frac{1}{2n^2}} \approx \sqrt{2} \) (since \( \frac{1}{2n^2} \) is very small).

Now, we approximate:

\[
\frac{1}{2} \left( \sqrt{a_1} + \sqrt{a_2} + \ldots + \sqrt{a_{119}} \right) \approx \frac{1}{2} \left( 119 \times \sqrt{2} \right)
= \frac{119 \sqrt{2}}{2}
\]

\[
= 59.5 \sqrt{2}
\]

### Final Answer

\[
\frac{1}{2} \left( \sqrt{a_1} + \sqrt{a_2} + \ldots + \sqrt{a_{119}} \right) \approx 59.5 \sqrt{2}
\]

### Additional Questions and Tips

1. How does rationalizing the denominator help in simplifying expressions?
2. Why is it useful to approximate sequences when \( n \) is large?
3. What are other methods to approximate square roots of expressions like \( \sqrt{2 - \frac{1}{2n^2}} \)?
4. Can we find a general expression for \( \sqrt{a_n} \) without approximation?
5. How do small terms affect the accuracy of approximations in series or sequences?

**Tip**: Always check the behavior of sequences as \( n \to \infty \) to see if terms approach a limit, simplifying calculations.

Learn how to evaluate the sequence {a_n} and calculate the sum of square roots for {a_1} to {a_{119}} using mathematical approximations and rationalization techniques. Suitable for advanced high school and college-level students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation