Problem 5 - Precalculus: Mathematics for Calculus

Question

Find the first four terms and the 100 th term of the sequence whose \(n\)th term is given. \(a_{n}=\frac{1}{2 n+1}\)

Step-by-Step Solution

Answer

The first four terms are \(\frac{1}{3}, \frac{1}{5}, \frac{1}{7}, \frac{1}{9}\), and the 100th term is \(\frac{1}{201}\).

Step 1: Find the first term

To find the first term of the sequence, substitute \(n = 1\) into the formula \(a_{n} = \frac{1}{2n + 1}\). This gives us \(a_{1} = \frac{1}{2(1) + 1} = \frac{1}{3}\).

Step 2: Find the second term

Substitute \(n = 2\) into the formula \(a_{n} = \frac{1}{2n + 1}\). This gives us \(a_{2} = \frac{1}{2(2) + 1} = \frac{1}{5}\).

Step 3: Find the third term

Substitute \(n = 3\) into the formula \(a_{n} = \frac{1}{2n + 1}\). This gives us \(a_{3} = \frac{1}{2(3) + 1} = \frac{1}{7}\).

Step 4: Find the fourth term

Substitute \(n = 4\) into the formula \(a_{n} = \frac{1}{2n + 1}\). This gives us \(a_{4} = \frac{1}{2(4) + 1} = \frac{1}{9}\).

Step 5: Find the 100th term

Substitute \(n = 100\) into the formula \(a_{n} = \frac{1}{2n + 1}\). This gives us \(a_{100} = \frac{1}{2(100) + 1} = \frac{1}{201}\).

Key Concepts

nth Term Formula

In mathematics, sequences are orderly collections of numbers, offering a unique pattern. The "nth term formula" is a powerful tool used to identify any specific term in the sequence without listing all previous terms. This formula, denoted as \(a_n\), gives you the freedom to find any term as long as you know its position in line
For instance, in the sequence characterized by the formula \(a_{n} = \frac{1}{2n + 1}\), substituting a number for \(n\) yields the desired result:

• Plug in \(n = 1\), and you get \(a_1 = \frac{1}{3}\), the very first term.
• Substitute \(n = 100\) to directly calculate \(a_{100} = \frac{1}{201}\), without explicitly computing the first 99 terms.

A general nth term formula let's you hop directly to any term with few simple calculations, saving valuable time and effort.

Arithmetic Sequences

Arithmetic sequences are a special type of sequence where the difference between consecutive terms remains constant. This difference is called the "common difference." However, not every sequence is arithmetic. A sequence is deemed arithmetic if it adheres to this rule of consistent difference. For example, if a sequence begins with 3, 5, 7, it's adding 2 each time—in other words, its common difference is 2.
The sequence defined by \(a_{n} = \frac{1}{2n + 1}\) does not exhibit this property. Each term is uniquely calculated using its formula, resulting in non-constant differences between terms. Hence, this sequence is not arithmetic but is still a valuable function to explore another exciting mathematical concept. Not all sequences need to fit the arithmetic mold to be useful or interesting.

Step-by-Step Problem Solving

Breaking down a problem into step-by-step solutions makes complex mathematical tasks understandable. By solving each piece one at a time, you go from confusion to clarity.
Take the sequence \(a_{n} = \frac{1}{2n + 1}\) as an example:

• Step 1: Find the first term by substituting \(n = 1\). Solution: \(a_1 = \frac{1}{3}\).
• Step 2: Substitute \(n = 2\) to calculate the second term. Solution: \(a_2 = \frac{1}{5}\).
• Step 3: For the third term, use \(n = 3\). Solution: \(a_3 = \frac{1}{7}\).
• Step 4: Find the fourth term with \(n = 4\). Solution: \(a_4 = \frac{1}{9}\).
• Step 5: Directly jump to the 100th term with \(n = 100\). Solution: \(a_{100} = \frac{1}{201}\).

Following each step methodically not only helps grasp the process but also builds confidence, enabling you to tackle more challenging sequences with ease. Problems become manageable and solutions emerge naturally from orderly progression.