Problem 13 - College Algebra with Modeling and Visualization

Question

For the given \(a_{n},\) calculate \(S_{5}.\) $$ a_{\mathrm{n}}=\frac{n}{n+1} $$

Step-by-Step Solution

Answer

The sum \(S_5\) is \(\frac{71}{20}\).

Step 1: Understand the problem

We are given the sequence \(a_n = \frac{n}{n+1}\). We need to find the sum of the first five terms, denoted as \(S_5\). This means adding together \(a_1, a_2, a_3, a_4,\) and \(a_5\).

Step 2: Calculate the first term \(a_1\)

Substitute \(n=1\) into the formula for \(a_n\): \[a_1 = \frac{1}{1+1} = \frac{1}{2}\] So, \(a_1 = \frac{1}{2}\).

Step 3: Calculate the second term \(a_2\)

Substitute \(n=2\) into the formula for \(a_n\): \[a_2 = \frac{2}{2+1} = \frac{2}{3}\] Thus, \(a_2 = \frac{2}{3}\).

Step 4: Calculate the third term \(a_3\)

Substitute \(n=3\) into the formula for \(a_n\): \[a_3 = \frac{3}{3+1} = \frac{3}{4}\] Thus, \(a_3 = \frac{3}{4}\).

Step 5: Calculate the fourth term \(a_4\)

Substitute \(n=4\) into the formula for \(a_n\): \[a_4 = \frac{4}{4+1} = \frac{4}{5}\] Thus, \(a_4 = \frac{4}{5}\).

Step 6: Calculate the fifth term \(a_5\)

Substitute \(n=5\) into the formula for \(a_n\): \[a_5 = \frac{5}{5+1} = \frac{5}{6}\] Thus, \(a_5 = \frac{5}{6}\).

Step 7: Calculate \(S_5\)

Now, sum the first five terms: \[S_5 = a_1 + a_2 + a_3 + a_4 + a_5\] Substitute the values we calculated: \[S_5 = \frac{1}{2} + \frac{2}{3} + \frac{3}{4} + \frac{4}{5} + \frac{5}{6}\]

Step 8: Find a common denominator and simplify

One common denominator for 2, 3, 4, 5, and 6 is 60. Convert each fraction: \[\frac{1}{2} = \frac{30}{60}, \quad \frac{2}{3} = \frac{40}{60}, \quad \frac{3}{4} = \frac{45}{60}, \quad \frac{4}{5} = \frac{48}{60}, \quad \frac{5}{6} = \frac{50}{60}\] Add them together: \[S_5 = \frac{30}{60} + \frac{40}{60} + \frac{45}{60} + \frac{48}{60} + \frac{50}{60} = \frac{213}{60}\]

Step 9: Simplify the fraction

Simplify the fraction \(\frac{213}{60}\) by finding the greatest common divisor (GCD) of 213 and 60. The GCD is 3, so dividing both numerator and denominator by 3 gives us: \[\frac{213\div3}{60\div3} = \frac{71}{20}\] Therefore, \(S_5\) is \(\frac{71}{20}\).

Key Concepts

Sequence Calculation

In this exercise, we begin with an arithmetic sequence given by the formula \( a_n = \frac{n}{n+1} \). A sequence is basically a list of numbers generated based on a specific rule or formula. Here, the job is to apply this formula for each term up to the fifth one, i.e., from \( a_1 \) to \( a_5 \).To calculate the terms:

• Substitute \( n \) with 1 to find \( a_1 \), giving \( \frac{1}{2} \).
• Continue with \( n=2 \) for \( a_2 \), leading to \( \frac{2}{3} \).
• Repeat this process until \( n=5 \) for \( a_5 \), which is \( \frac{5}{6} \).

This approach is straightforward as you methodically replace \( n \) with the term number to produce the sequence values step-by-step. Understanding sequence calculations helps with identifying patterns and solving complex sequence problems with ease.

Sum of Series

Once individual sequence numbers from \( a_1 \) to \( a_5 \) are identified, the next step is adding them all to find the sum \( S_5 \). The sum of the series for the first five terms is represented as:\[ S_5 = a_1 + a_2 + a_3 + a_4 + a_5 \]Replacing each \( a_n \) value with its computed fraction gives us:\[ S_5 = \frac{1}{2} + \frac{2}{3} + \frac{3}{4} + \frac{4}{5} + \frac{5}{6} \]Adding up fractions involves a little finesse, which we handle in the next section. Calculating the sum of series like this can appear daunting at first. However, tackling it step-by-step makes the process clearer and more manageable. Summing up helps you see the overall growth or total values covered by the sequence up to a particular point.

Fraction Simplification

Adding fractions involves combining them into a single simplified expression. The essential step here is to find a common denominator. A common denominator for 2, 3, 4, 5, and 6 is 60, which allows us to easily add the fractions.Let's convert each:

• \( \frac{1}{2} = \frac{30}{60} \)
• \( \frac{2}{3} = \frac{40}{60} \)
• \( \frac{3}{4} = \frac{45}{60} \)
• \( \frac{4}{5} = \frac{48}{60} \)
• \( \frac{5}{6} = \frac{50}{60} \)

Add them together:\[ S_5 = \frac{213}{60} \]Finding the greatest common divisor (GCD) of 213 and 60, which is 3, simplifies the fraction to \( \frac{71}{20} \). Simplification is crucial in mathematical problem-solving. It enables clearer interpretations and applications, providing a neat, reduced expression that reflects the total value accurately.