Problem 18 - Calculus for Biology and Medicine

Question

Write each sum in sigma notation. $$ \frac{3}{5}+\frac{4}{6}+\frac{5}{7}+\frac{6}{8}+\frac{7}{9} $$

Step-by-Step Solution

Answer

\( \sum_{n=1}^{5} \frac{n+2}{n+4} \)

Step 1: Identify the Pattern

First, observe the given sequence of terms: \( \frac{3}{5}, \frac{4}{6}, \frac{5}{7}, \frac{6}{8}, \frac{7}{9} \). Notice that the numerator increases by 1 each time starting from 3 and the denominator increases by 1 each time starting from 5.

Step 2: Determine the General Term

From the pattern, define the general term of the sequence. For the \( n^{th} \) term: The numerator increases by 1 starting from 3, can be written as \( n + 2 \). The denominator also increases by 1 starting from 5, can be written as \( n + 4 \). Hence, the general term for the \( n^{th} \) term is \( \frac{n+2}{n+4} \).

Step 3: Specify the Index of Summation

Identify the range of the index. The sequence is given from \( n = 1 \) to \( n = 5 \). This is determined by the fact that there are 5 terms in the original expression, starting from the first term \( \frac{3}{5} \) when \( n = 1 \), to the fifth term \( \frac{7}{9} \) when \( n = 5 \).

Step 4: Write in Sigma Notation

Using the results from Steps 1, 2, and 3, write the sum in sigma notation. The sum is: \[ \sum_{n=1}^{5} \frac{n+2}{n+4} \]

Key Concepts

Calculus

Calculus is fundamental in mathematics, particularly when dealing with sequences and series like the one in our problem. Calculus explores concepts such as limits, functions, derivatives, and integrals. In the broad scope of calculus, we often use sigma notation (\( \Sigma \)) to succinctly express the summation of sequences. This becomes especially useful when the sequence follows a specific pattern that can be generalized, as shown in the exercise.

Understanding calculus allows us to evaluate infinite sums, grasp continuous growth or decay, and handle various functions and their behaviors. In this particular exercise, being able to write a sequence in sigma notation is a stepping stone to more advanced calculus concepts such as finding the sum of infinite series or solving differential equations. While this problem deals with a finite series, the mathematical tools developed in calculus help create a bridge to understanding more complex mathematical ideas.

Sequence Pattern

Identifying sequence patterns involves recognizing a repeated or consistent theme in a list of numbers. In the original sequence, we see the pattern by observing that each term's numerator and denominator follow a consistent change.

Here’s what to look for when identifying patterns in a sequence:

• Check how each term changes from one to the next.
• Look for both numerators and denominators separately.
• Assess whether there is a consistent operation (addition, multiplication, etc.).

In our case, the numerator starts at 3 and each subsequent term increases by 1. Similarly, the denominator starts at 5 with the same incremental pattern, increasing by 1 each time. By noticing these patterns, the expression can be generalized to a function of \( n \), such as \( \frac{n+2}{n+4} \), allowing for easy expression in sigma notation as shown in the solution.

Understanding sequence patterns is crucial in mathematics as it aids in constructing formulas that can predict or extend the sequence beyond its given terms, paving the way for analyzing larger datasets efficiently.

Index of Summation

The index of summation is a critical component when expressing sequences in sigma notation. It tells us where to start and stop our summation. In our exercise, the index of summation is \( n \), which runs from 1 to 5. This specifies that we sum up the terms as \( n \) varies over the integers 1 through 5.

When determining the index of summation, follow these steps:

• Identify the position of the first term in the sequence.
• Identify the position of the last term.
• Count how many terms are there in total.

The start and end values convey how many terms are included in the sum. In the given problem, since there are 5 terms span from \( n=1 \) to \( n=5 \). Thus, the sum is expressed from 1 to 5.

Mastering the concept of the index of summation is fundamental in working with series as it helps define the limits of operations and is an essential component in integrating more complex calculus tasks.