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Chapter 9: Problem 30

\(y_{n}-2 y_{n-1}=8 ; y_{0}=-1\)

Short Answer

Expert verified
The terms are:\( y_0 = -1 \)\( y_1 = 6 \)\( y_2 = 20 \)\( y_3 = 48 \)

Step by step solution

01

Identify the Initial Condition and Equation

The given recursive equation is: \[ y_{n} - 2y_{n-1} = 8 \] The initial condition is: \[ y_0 = -1 \] This will help in finding the first few terms.
02

Calculate the First Term

Using the initial condition: \[ y_0 = -1 \] To find the next term, substitute \( n = 1 \) into the equation: \[ y_1 - 2y_0 = 8 \] Replace \( y_0 \) with -1: \[ y_1 - 2(-1) = 8 \] Simplify: \[ y_1 + 2 = 8 \] Isolate \( y_1 \): \[ y_1 = 6 \]
03

Calculate the Second Term

To find \( y_2 \), substitute \( n = 2 \) into the equation: \[ y_2 - 2y_1 = 8 \] Replace \( y_1\) with 6: \[ y_2 - 2(6) = 8 \] Simplify: \[ y_2 - 12 = 8 \] Isolate \( y_2 \): \[ y_2 = 20 \]
04

Calculate the Third Term

To find \( y_3 \), substitute \( n = 3 \) into the equation: \[ y_3 - 2y_2 = 8 \] Replace \( y_2 \) with 20: \[ y_3 - 2(20) = 8 \] Simplify: \[ y_3 - 40 = 8 \] Isolate \( y_3 \): \[ y_3 = 48 \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Initial Condition
An initial condition in a recursive sequence helps to kickstart the sequence. It provides the starting value, or the very first term from which all other terms are derived.

The given problem states the initial condition as: \( y_{0} = -1 \). This means that the very first term, often referred to as the zeroth term, in the sequence is -1.

Initial conditions are crucial because without them, we wouldn't have a reference point to start calculating subsequent terms. Essentially, they anchor the entire recursive process.
First Term
The first term in a recursive sequence is derivable from the initial condition and the recursive equation.

Given the initial condition \( y_{0} = -1 \), you can calculate the first term \( y_{1} \) using the recursive equation:

\( y_{n} - 2y_{n-1} = 8 \)

Substitute \( n = 1 \) into the equation:
\[ y_{1} - 2y_{0} = 8 \]
Using \( y_{0} = -1 \), it becomes:
\[ y_{1} - 2(-1) = 8 \]
Simplify it to find:
\[ y_{1} = 6 \]

So, the first term in the sequence is 6. This step-by-step calculation shows how essential the initial condition is to find the first term.
Recursive Equation
A recursive equation expresses each term of a sequence in terms of its preceding term(s). In this problem, the recursive equation is:
\[ y_{n} - 2 y_{n-1} = 8 \]

Here's how it works:

  • To find \( y_{2} \), substitute \( n = 2 \) into the equation:
    \[ y_{2} - 2 y_{1} = 8 \]Replace \( y_{1} \) with 6:
    \[ y_{2} - 2(6) = 8 \]
    Simplify:
    \[ y_{2} = 20 \]
  • To find \( y_{3} \), substitute \( n = 3 \):
    \[ y_{3} - 2 y_{2} = 8 \]
    Replace \( y_{2} \) with 20:
    \[ y_{3} - 2(20) = 8 \]
    Get:
    \[ y_{3} = 48 \]

Following this method, one can keep calculating further terms in the sequence. The recursive equation is the rule spelling out the relationship between consecutive terms and allows the sequence to be extended indefinitely.

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