Problem 39 - Algebra and Trigonometry

Question

\(37-40\) . Find the first four partial sums and the nth partial sum of the sequence \(a_{m}\) $$ a_{n}=\sqrt{n}-\sqrt{n+1} $$

Step-by-Step Solution

Answer

The nth partial sum is \(S_n = \sqrt{1} - \sqrt{n+1}\).

Step 1: Calculate the First Term

For the first partial sum, substitute \(n = 1\) into the sequence formula \(a_n = \sqrt{n} - \sqrt{n+1}\). This gives the first term \(a_1 = \sqrt{1} - \sqrt{2}\).

Step 2: Calculate the Second Partial Sum

To find the second partial sum, calculate the first two terms and add them. Using the formula, find \(a_2 = \sqrt{2} - \sqrt{3}\). The second partial sum is \(S_2 = a_1 + a_2 = (\sqrt{1} - \sqrt{2}) + (\sqrt{2} - \sqrt{3})\).

Step 3: Calculate the Third Partial Sum

Calculate \(a_3 = \sqrt{3} - \sqrt{4}\). Add the first three terms: \(S_3 = a_1 + a_2 + a_3 = (\sqrt{1} - \sqrt{2}) + (\sqrt{2} - \sqrt{3}) + (\sqrt{3} - \sqrt{4})\).

Step 4: Calculate the Fourth Partial Sum

Calculate \(a_4 = \sqrt{4} - \sqrt{5}\). Add together all four terms to get \(S_4 = a_1 + a_2 + a_3 + a_4 = (\sqrt{1} - \sqrt{2}) + (\sqrt{2} - \sqrt{3}) + (\sqrt{3} - \sqrt{4}) + (\sqrt{4} - \sqrt{5})\).

Step 5: Find the General nth Partial Sum

The nth partial sum \(S_n\) is the sum of the first \(n\) terms of the sequence, which telescopes to \(S_n = \sqrt{1} - \sqrt{n+1}\). Notice how intermediate terms cancel out.

Key Concepts

Sequence

A sequence is a structured, ordered list of numbers that follow a specific rule. In the sequence we're dealing with here, the rule is defined by the formula \(a_n = \sqrt{n} - \sqrt{n+1}\). This formula tells us how to calculate each term in the sequence based on the position, \(n\), of the term.

For example:

• When \(n = 1\), the term is \(a_1 = \sqrt{1} - \sqrt{2}\).
• For \(n = 2\), the term becomes \(a_2 = \sqrt{2} - \sqrt{3}\).
• This continues indefinitely as long as you have an \(n\) value to plug into the formula.

Observing a sequence helps us identify patterns or properties that are crucial in mathematical analysis. Each element is dependent on the \(n\) value, providing a unique perspective for each term in our ordered list.

Telescoping Series

A telescoping series is a special kind of series where many terms cancel each other out in a sequence of additions and subtractions. This makes it easier to find the sum of the series.

In our given sequence \(a_n = \sqrt{n} - \sqrt{n+1}\), when we add up the terms for partial sums, the intermediate terms cancel out. For instance:

• The first two terms, \((\sqrt{1} - \sqrt{2}) + (\sqrt{2} - \sqrt{3})\), reduce to \(\sqrt{1} - \sqrt{3}\).
• Similarly, adding more terms continues this pattern, leaving only the first and the last terms.

Thus, telescoping is a powerful characteristic that simplifies complex series, leading directly to the final result without calculating each term individually.

Series Summation

Series summation involves adding the terms of a sequence together. In many mathematical problems, we are interested in figuring out the total sum of a sequence of numbers.

For the sequence \(a_n = \sqrt{n} - \sqrt{n+1}\), when we sum over the first few terms, we realize that most terms cancel out due to the telescoping nature of the series. For instance:

• To find the first four partial sums, we calculate and add each term like \(S_4 = (\sqrt{1} - \sqrt{2}) + (\sqrt{2} - \sqrt{3}) + (\sqrt{3} - \sqrt{4}) + (\sqrt{4} - \sqrt{5})\).
• After simplifying, the result dramatically simplifies due to the telescoping cancellation, giving \(\sqrt{1} - \sqrt{5}\).

This illustrates that series summation isn't just about adding terms, but also about recognizing patterns and simplifications.

nth Partial Sum

The nth partial sum is crucial when analyzing sequences and series. It represents the total sum of the first \(n\) terms of the sequence. This acts like a snapshot of the series at a certain point.

The concept is particularly significant in our exercise where the nth partial sum \(S_n = \sqrt{1} - \sqrt{n+1}\) demonstrates how intermediate terms cancel. This nth partial sum outlines the effectiveness of telescoping as:

• The terms \(\sqrt{n} - \sqrt{n+1}\) cancel interim elements, ensuring the nth partial sum is simple and straightforward to compute.
• It provides a useful insight, showing that for any choice of \(n\), the process and calculus become straightforward.

Understanding the nth partial sum empowers learners to assess series comprehensively, especially when facing complex sequences in mathematical contexts.