Problem 42 - Algebra and Trigonometry

Question

A sequence is defined recursively. List the first five terms. \(a_{1}=-2 ; \quad a_{n}=n+3 a_{n-1}\)

Step-by-Step Solution

Answer

The first five terms are: -2, -4, -9, -23, -64.

Step 1 - Identify the first term

The first term of the sequence is given directly: \[ a_{1} = -2 \]

Step 2 - Find the second term

Use the recursive formula to find the second term: \[ a_{2} = 2 + 3a_{1} \] Substitute the value of \( a_{1} \): \[ a_{2} = 2 + 3(-2) = 2 - 6 = -4 \]

Step 3 - Calculate the third term

Use the recursive formula to find the third term: \[ a_{3} = 3 + 3a_{2} \] Substitute the value of \( a_{2} \): \[ a_{3} = 3 + 3(-4) = 3 - 12 = -9 \]

Step 4 - Determine the fourth term

Use the recursive formula to find the fourth term: \[ a_{4} = 4 + 3a_{3} \] Substitute the value of \( a_{3} \): \[ a_{4} = 4 + 3(-9) = 4 - 27 = -23 \]

Step 5 - Establish the fifth term

Use the recursive formula to find the fifth term: \[ a_{5} = 5 + 3a_{4} \] Substitute the value of \( a_{4} \): \[ a_{5} = 5 + 3(-23) = 5 - 69 = -64 \]

Key Concepts

Sequence Definition

A sequence is a list of numbers arranged in a specific order. Each number in the sequence is called a term. Sequences can be defined in different ways, such as explicitly by a formula or recursively. An explicit definition gives a direct formula for any term based on its position, while a recursive definition specifies each term based on previous terms. Recursive sequences are especially useful in modeling real-world problems where each term depends on prior values.

Recursive Formula

A recursive formula provides a way to find each term of a sequence by relating it to previous terms. For a sequence \(a_n\), the recursive formula typically has two parts:

• Initial condition: one or more starting values.
• Recurrence relation: a rule that defines \(a_n\) based on earlier terms in the sequence.

In our exercise, the sequence begins with \(a_1 = -2\), and each subsequent term \(a_n\) is calculated using the relation \ a_n = n + 3a_{n-1} \.

Term Calculation

Calculating the terms of a recursive sequence involves applying the recursive formula step by step. Let's break down the steps:

• **First Term:** It's given directly as \(a_1 = -2\).
• **Second Term:** Use the formula \(a_2 = 2 + 3a_1\). Substituting \(a_1 = -2\), we get \ a_2 = 2 + 3(-2) = -4 \.
• **Third Term:** \(a_3 = 3 + 3a_2\). Using \ a_2 = -4 \, we find \ a_3 = 3 + 3(-4) = -9 \.
• **Fourth Term:** \(a_4 = 4 + 3a_3\). Given \ a_3 = -9 \, \ a_4 = 4 + 3(-9) = -23 \.
• **Fifth Term:** \ a_5 = 5 + 3a_4 \. Substituting \ a_4 = -23 \, we get \ a_5 = 5 + 3(-23) = -64 \.

Algebra

Algebra plays a crucial role in working with recursive sequences. By understanding algebraic operations, you can simplify and solve the recurrence relations. Here are some important points:

• **Substitution:** This is used to replace a term with its known value to find the next term.
• **Simplification:** Combining like terms and performing arithmetic operations correctly ensures accurate solutions.
• **Identifying Patterns:** Recognize patterns in sequences which can sometimes help in finding shortcuts or validating calculations.

In our exercise, we repeatedly used substitution and simplification to find each subsequent term in the sequence. Mastery in basic algebra makes handling recursive sequences far easier.