Problem 10 - Algebra and Trigonometry

Question

write the first four terms of each sequence whose general term is given. $$ a_{n}=\frac{3 n}{n+5} $$

Step-by-Step Solution

Answer

The first four terms of the sequence are 0.5, 0.857, 1.125, 1.333.

Step 1: Substitute \(n\) with 1

Substitute the variable \(n\) in the formula with 1. So we get: \(a_{1}=\frac{3(1)}{1+5}=\frac{3}{6} = 0.5\)

Step 2: Substitute \(n\) with 2

Substitute the variable \(n\) in the formula with 2. So we get: \(a_{2}=\frac{3(2)}{2+5}=\frac{6}{7} \approx 0.857\)

Step 3: Substitute \(n\) with 3

Substitute the variable \(n\) in the formula with 3. So we get: \(a_{3}=\frac{3(3)}{3+5}=\frac{9}{8} = 1.125\)

Step 4: Substitute \(n\) with 4

Substitute the variable \(n\) in the formula with 4. So we get: \(a_{4}=\frac{3(4)}{4+5}=\frac{12}{9} = 1.333\)

Key Concepts

General Term

In the world of mathematical sequences, the general term is essentially a formula that expresses the terms of the sequence based on their position, denoted by the index \( n \). For instance, in the sequence expressed by \( a_{n} = \frac{3n}{n+5} \), \( n \) refers to the position of the term within the sequence. This allows us to generate any specific term by plugging in the value of \( n \). Understanding the general term is key to unraveling the pattern a sequence follows. It acts like a road map that tells you how to compute any term in the series, saving you from having to start from scratch each time.

Substitution

Substitution is a straightforward but vital mathematical tool used to find specific values for variables by replacing them with known numbers. In the context of sequences, substitution enables us to determine the various terms by systematically replacing \( n \) with consecutive integers. For instance, in the given sequence formula \( a_{n} = \frac{3n}{n+5} \), when we substitute \( n \) with 1,2,3, and 4, we find the first four terms of the sequence. This process illuminates how changing the position index \( n \) alters the term's value. Practicing substitution sharpens your ability to solve sequences effortlessly.

Mathematical Sequences

Mathematical sequences are orderly lists of numbers defined by a particular formula or pattern. In a sequence, each term is derived from a specific position or index \( n \). These sequences can be finite or infinite. The beauty of sequences is in their predictability and the ease with which one can ascertain any term using the sequence's general term formula. For example, the sequence derived from \( a_{n} = \frac{3n}{n+5} \) shows a progression where each term is linked to the next by a simple arithmetic operation. Sequences commonly encounter in math include arithmetic and geometric ones, each with unique defining properties and formulas.

Arithmetic Evaluation

Arithmetic evaluation involves performing basic operations such as addition, subtraction, multiplication, and division to arrive at a numeric result. When evaluating a sequence term like \( a_{n} = \frac{3n}{n+5} \), arithmetic evaluation comes into play as you substitute the index \( n \) and carry out the calculations to find the value of each term. For example, \( a_1 = \frac{3(1)}{1+5} = \frac{3}{6} = 0.5 \) is an arithmetic evaluation process. Each term's calculation in a sequence helps reinforce arithmetic skills while also demonstrating how formulas and operations are practically applied to generate useful information.