Problem 5 - Precalculus Mathematics for Calculus

Question

Find the first four terms and the 100th term of the sequence. $$a_{n}=\frac{1}{n+1}$$

Step-by-Step Solution

Answer

The first four terms are \( \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \frac{1}{5} \), and the 100th term is \( \frac{1}{101} \).

Step 1: Identify the general formula

The general formula for the sequence is given as \( a_n = \frac{1}{n+1} \). This formula will be used to find the terms of the sequence by substituting the appropriate values of \( n \).

Step 2: Calculate the first term

Substitute \( n = 1 \) into the formula: \[ a_1 = \frac{1}{1+1} = \frac{1}{2} \] Thus, the first term is \( \frac{1}{2} \).

Step 3: Calculate the second term

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

Step 4: Calculate the third term

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

Step 5: Calculate the fourth term

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

Step 6: Calculate the 100th term

Substitute \( n = 100 \) into the formula: \[ a_{100} = \frac{1}{100+1} = \frac{1}{101} \] Thus, the 100th term is \( \frac{1}{101} \).

Key Concepts

General Formula

In mathematics, a sequence is an ordered list of elements, often numbers. The general formula serves as the blueprint for finding any term in a sequence. This formula is like a guiding map that tells you how to compute each term. Here, the general formula for our sequence is \( a_n = \frac{1}{n+1} \), where \( n \) represents the position of the term within the sequence.

If you know the general formula, you're already halfway to determining any term you want! By plugging in different values of \( n \) into the formula, you can calculate each term easily. The beauty here is that you don't need to memorize each result individually; you only need this single expression to find any term.

Terms of the Sequence

Terms refer to individual numbers or elements in a sequence. Based on the general formula, you can generate each term by substituting \( n \) with its position. For example:

• When \( n = 1 \), the first term \( a_1 \) is \( \frac{1}{2} \).
• When \( n = 2 \), the second term \( a_2 \) is \( \frac{1}{3} \).
• When \( n = 3 \), the third term \( a_3 \) is \( \frac{1}{4} \).
• When \( n = 4 \), the fourth term \( a_4 \) is \( \frac{1}{5} \).

Each of these terms is obtained by simply following the rule set by the general formula.

Remember, a sequence can have countless terms, which means endless possibilities. Using the sequence's rule, you can discover terms far in the future of the sequence, such as the 100th term! All you need to do is adjust \( n \) to obtain your desired term's position.

Substitution Method

The substitution method is essentially replacing a variable in a formula with a specific value. Here, we take the general formula \( a_n = \frac{1}{n+1} \) and substitute \( n \) with the term number we want to find.

For instance, to find the 100th term, substitute \( n = 100 \):

• \( a_{100} = \frac{1}{100+1} = \frac{1}{101} \)

This method is a straightforward approach. It's just a matter of plugging the right numbers into the formula. This technique ensures precision because it strictly adheres to the given sequence rule.

By repeatedly applying the substitution method, you can pinpoint any position within the sequence. It's a simple yet powerful tool in sequences that grants you access to every term within your reach.