Problem 17 - Algebra and Trigonometry

Question

List the first five terms of each sequence. \(\left\\{a_{n}\right\\}=\left\\{\frac{n}{n+2}\right\\}\)

Step-by-Step Solution

Answer

The first five terms are \(\frac{1}{3}\), \(\frac{1}{2}\), \(\frac{3}{5}\), \(\frac{2}{3}\), and \(\frac{5}{7}\).

Step 1 - Identify the General Term

The general term of the sequence is given by \(a_{n} = \frac{n}{n+2}\). This formula will be used to find each of the first five terms.

Step 2 - Calculate the First Term

To find the first term, substitute \(n = 1\) into the general term: \(a_1 = \frac{1}{1+2} = \frac{1}{3}\).

Step 3 - Calculate the Second Term

To find the second term, substitute \(n = 2\) into the general term: \(a_2 = \frac{2}{2+2} = \frac{2}{4} = \frac{1}{2}\).

Step 4 - Calculate the Third Term

To find the third term, substitute \(n = 3\) into the general term: \(a_3 = \frac{3}{3+2} = \frac{3}{5}\).

Step 5 - Calculate the Fourth Term

To find the fourth term, substitute \(n = 4\) into the general term: \(a_4 = \frac{4}{4+2} = \frac{4}{6} = \frac{2}{3}\).

Step 6 - Calculate the Fifth Term

To find the fifth term, substitute \(n = 5\) into the general term: \(a_5 = \frac{5}{5+2} = \frac{5}{7}\).

Key Concepts

headline of the respective core concept

Mathematical sequences can often seem confusing, but they can be broken down into simple parts. When working with sequences, understanding the general term is crucial. The general term is a formula that gives the value of each term in the sequence based on its position (n). For instance, in the sequence where \(a_{n} = \frac{n}{n+2}\), the general term \(\frac{n}{n+2}\) tells us how to calculate any term by plugging in the value of n.

headline of the respective core concept

Now that we know the general term, we can calculate individual terms. Each term in a sequence is found by substituting the position number into the general term formula. Let's go through this step-by-step:To find the first term, substitute \(n = 1\): \(a_1 = \frac{1}{1+2} = \frac{1}{3}\).To find the second term, substitute \(n = 2\): \(a_2 = \frac{2}{2+2} = \frac{2}{4} = \frac{1}{2}\).To find the third term, substitute \(n = 3\): \(a_3 = \frac{3}{3+2} = \frac{3}{5}\).Continue this process to find more terms in the sequence.

headline of the respective core concept

Sometimes, the terms in a sequence are best represented as fractions. In fractional representation, we break down terms into their simplest form. For example, in our sequence, \(\a_n\) terms often need simplifying. Observe how: \(\frac{2}{4}\) is simplified to \(\frac{1}{2}\). Similarly, \(\frac{4}{6}\) simplifies to \(\frac{2}{3}\). This makes the sequence easier to understand and work with.