√n+1 -√n write the first fout elements of the sequence and determine whether it is convergent or divergent (substitute n to 1, 2, 3 and 4)

Let's consider the expression \( \sqrt{n+1} - \sqrt{n} \) and compute the first four terms of the sequence by substituting \( n = 1, 2, 3, 4 \):

1. For \( n = 1 \):
\[
\sqrt{1+1} - \sqrt{1} = \sqrt{2} - \sqrt{1} = \sqrt{2} - 1 \approx 1.414 - 1 = 0.414
\]

2. For \( n = 2 \):
\[
\sqrt{2+1} - \sqrt{2} = \sqrt{3} - \sqrt{2} \approx 1.732 - 1.414 = 0.318
\]

3. For \( n = 3 \):
\[
\sqrt{3+1} - \sqrt{3} = \sqrt{4} - \sqrt{3} = 2 - 1.732 = 0.268
\]

4. For \( n = 4 \):
\[
\sqrt{4+1} - \sqrt{4} = \sqrt{5} - \sqrt{4} \approx 2.236 - 2 = 0.236
\]

Thus, the first four terms of the sequence are approximately:
\[
0.414, \, 0.318, \, 0.268, \, 0.236
\]

### Convergence or Divergence:
As \( n \to \infty \), both \( \sqrt{n+1} \) and \( \sqrt{n} \) grow large, but the difference between them becomes smaller. More precisely:

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

As \( n \to \infty \), the denominator \( \sqrt{n+1} + \sqrt{n} \) grows large, making the entire expression approach 0. Therefore, the sequence converges to 0.

### Conclusion:
The sequence is **convergent**.

Would you like further details or have any questions?

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Here are 5 related questions to expand your understanding:

1. What is the general form of the limit for sequences involving square roots?
2. How can you prove the convergence of a sequence using mathematical limits?
3. What happens if you try this with a different form, like \( \sqrt{n+2} - \sqrt{n} \)?
4. How would the sequence behave if \( n \) starts at 0 instead of 1?
5. How can we approximate the difference \( \sqrt{n+1} - \sqrt{n} \) for large \( n \)?

**Tip:** In sequences involving square roots, rationalizing the expression often helps in determining convergence properties!

√n+1 -√n write the first four elements of the sequence and determine whether it is convergent or divergent (substitute n to 1, 2, 3 and 4)

Explore the sequence √n+1 - √n by calculating its first four elements and analyzing its convergence. Learn key concepts in sequences, square roots, and limits, and understand why the sequence converges to 0 as n approaches infinity. Suitable for students in Grades 10-12.

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