Problem 5 - Algebra and Trigonometry

Question

\(3-12\) . Find the first four terms and the 100 th term of the sequence. $$ a_{n}=\frac{1}{n+1} $$

Step-by-Step Solution

Answer

First four terms: \( \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \frac{1}{5} \); 100th term: \( \frac{1}{101} \).

Step 1: Identify the Sequence Formula

The given sequence is defined by the formula \( a_n = \frac{1}{n+1} \). This formula will be used to calculate any term in the sequence.

Step 2: Calculate the First Term

To find the first term \( a_1 \), substitute \( n = 1 \) into the formula: \[ a_1 = \frac{1}{1+1} = \frac{1}{2} \] So, the first term is \( \frac{1}{2} \).

Step 3: Calculate the Second Term

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

Step 4: Calculate the Third Term

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

Step 5: Calculate the Fourth Term

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

Step 6: Calculate the 100th Term

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

Key Concepts

Sequence formula

A sequence formula is a mathematical expression that defines the terms of a sequence. In this exercise, we use the formula \( a_n = \frac{1}{n+1} \), where each term \( a_n \) is calculated by plugging a whole number \( n \) into the formula. This simple rule helps in finding any term of the sequence by only adjusting the value of \( n \).The formula provides a systematic approach to generate an entire sequence. Once you have the formula, you can easily explore a sequence further. For example, by substituting different values of \( n \), you can find terms like \( a_1, a_2, a_3, \) and so on. Sequence formulas make patterns within sequences visible and help identify their nature, making them invaluable tools in arithmetic sequences.

Term calculation

Term calculation involves using the sequence formula to find specific terms within a sequence. For example, with the formula \( a_n = \frac{1}{n+1} \), substituting \( n = 1 \) gives the first term \( a_1 = \frac{1}{2} \).Calculating terms involves three easy steps:

• Substitute the desired term number (\( n \)) into the sequence formula.
• Simplify the expression, following standard arithmetic rules.
• Identify the term based on your calculation.

This process is repeated for each term you are interested in. Whether it is the first, second, or hundredth term, with the correct substitution and simplification, you can compute any required term in the sequence. Term calculation is fundamental for understanding sequences as it aids in uncovering the properties of any sequence.

Arithmetic progression

Arithmetic progression (AP) refers to a sequence where each term after the first is produced by adding a fixed number, called the common difference, to the previous term. However, the sequence in this exercise does not demonstrate an arithmetic progression because there is no constant difference between successive terms.In an arithmetic progression such as \( a, a+d, a+2d, \ldots \), the common difference \( d \) is the same between every two consecutive terms. This exercise addresses the concept of following a specific formula rather than adding a common difference. While the sequence \( a_n = \frac{1}{n+1} \) follows a formulaic pattern, its approach differs from arithmetic progressions. Understanding this distinction helps in recognizing various sequence types and their specific characteristics.