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A sequence is an ordered list of numbers following a particular pattern or rule. Each number in a sequence is called a term. For example, consider the sequence defined as \(u_n = \sqrt{n + 1} - \sqrt{n}\). This sequence follows a pattern where each term is calculated by subtracting the square root of \(n\) from the square root of \(n+1\).

To get a better grasp of how this works, it's helpful to look at the first few terms:

• The first term is \(u_1 = \sqrt{1 + 1} - \sqrt{1} = \sqrt{2} - 1\).
• The second term is \(u_2 = \sqrt{2 + 1} - \sqrt{2} = \sqrt{3} - \sqrt{2}\).
• The third term is \(u_3 = \sqrt{3 + 1} - \sqrt{3} = 2 - \sqrt{3}\).
• Finally, the fourth term is \(u_4 = \sqrt{4 + 1} - 2 = \sqrt{5} - 2\).

Understanding sequences is key in mathematics because it forms the basis for analyzing patterns, progressions, and series.

A telescoping series is a special type of series where many terms cancel out when added together. This makes it easier to find the sum of the series. In the given exercise, the sequence \(u_n = \sqrt{n + 1} - \sqrt{n}\) forms a telescoping series. Let's see how this works.

When you sum up the sequence from 1 to \(n\), the terms begin to cancel each other. For example:

• Among the terms of \(u_1 = \sqrt{2} - \sqrt{1}\), \(u_2 = \sqrt{3} - \sqrt{2}\), and \(u_3 = 2 - \sqrt{3}\), the middle parts \(\sqrt{2}\) and \(\sqrt{3}\) cancel out.

This cancellation process continues, leaving only the first and last terms:
\[S_n = (\sqrt{2} - 1) + (\sqrt{3} - \sqrt{2}) + (2 - \sqrt{3}) + \ldots + (\sqrt{n+1} - \sqrt{n})\]After the cancellation, it simplifies to:
\[S_n = \sqrt{n+1} - 1\]Telescoping series are useful because they simplify long sums and make it easier to find partial sums.

The general expression for a series gives you a formula to compute any term or sum within the series without calculating each term individually. In the context of our sequence \(u_n = \sqrt{n+1} - \sqrt{n}\), the goal is to find a formula for the \(n\) th partial sum, \(S_n\).

With telescoping, we derived that the \(n\) th partial sum \(S_n\) can be expressed as:

• \(S_n = \sqrt{n+1} - 1\)

This shows that regardless of which number \(n\) you choose, the sum of the first \(n\) terms will always be \(\sqrt{n+1} - 1\). Hence, the general expression provides a simple shortcut for computation.

Using a general expression enhances efficiency in mathematical calculations and helps in quickly analyzing the behavior of the sequence as \(n\) becomes larger. This approach negates the need to calculate step-by-step, saving time and reducing error in processes involving large numbers of terms.