Problem 95

Question

The sequence $\mid a_{n}$ \\} is recursively defined. Compute $a_{n}$ for $n=1,2, \ldots, 5$ $$ a_{n+1}=1+2 a_{n}, a_{0}=0 $$

Step-by-Step Solution

Verified

Answer

The terms are $ a_1 = 1 $, $ a_2 = 3 $, $ a_3 = 7 $, $ a_4 = 15 $, $ a_5 = 31 $.

1Step 1: Understand the Recursive Formula

The sequence is defined by the formula $ a_{n+1} = 1 + 2a_n $ and the initial condition $ a_0 = 0 $. The goal is to compute $ a_n $ for $ n = 1, 2, \, \ldots, \, 5 $.

2Step 2: Compute the First Term

To find $ a_1 $, use the formula with $ n = 0 $: \[ a_{1} = 1 + 2a_0 = 1 + 2(0) = 1. \]

3Step 3: Compute the Second Term

To find $ a_2 $, use the formula with $ n = 1 $: \[ a_2 = 1 + 2a_1 = 1 + 2(1) = 1 + 2 = 3. \]

4Step 4: Compute the Third Term

To find $ a_3 $, use the formula with $ n = 2 $: \[ a_3 = 1 + 2a_2 = 1 + 2(3) = 1 + 6 = 7. \]

5Step 5: Compute the Fourth Term

To find $ a_4 $, use the formula with $ n = 3 $: \[ a_4 = 1 + 2a_3 = 1 + 2(7) = 1 + 14 = 15. \]

6Step 6: Compute the Fifth Term

To find $ a_5 $, use the formula with $ n = 4 $: \[ a_5 = 1 + 2a_4 = 1 + 2(15) = 1 + 30 = 31. \]

Key Concepts

Recurrence RelationInitial ConditionsSequence Computation

Recurrence Relation

A recurrence relation is a mathematical expression that defines a sequence where each term is a function of its preceding terms. In simpler terms, it tells us how to get the next number in a series from the numbers we already know.
In the exercise provided, the sequence is described by the recurrence relation:

$ a_{n+1} = 1 + 2a_n $

This means the next term, $ a_{n+1} $, can be calculated by taking twice the current term, $ a_n $, and then adding 1.
This pattern allows us to generate subsequent numbers as long as we know the starting point. Understanding the relation is crucial as it acts as an instruction manual for building the sequence.

Initial Conditions

The initial condition in a sequence is the starting point or first value from which all other values are calculated. Initial conditions are essential because they anchor the sequence, providing a known value to plug into the recurrence relation.
In our sequence, the initial condition given is:

$ a_0 = 0 $

This tells us that the value of the sequence when $ n = 0 $ is zero. This first term is essential because every following term is derived from it using the recurrence relation.
Without this, we wouldn't have a fixed point to begin our calculations, rendering the recurrence relation useless for producing subsequent terms.

Sequence Computation

Sequence computation involves using the recurrence relation and initial conditions to determine the terms of the sequence. Let's walk through computing the first five terms in our sequence step by step:

For $ n = 1 $, substitute $ a_0 = 0 $ into the relation: $ a_1 = 1 + 2(0) = 1 $.
For $ n = 2 $, use the value of $ a_1 = 1 $: $ a_2 = 1 + 2(1) = 3 $.
Next, for $ n = 3 $, take $ a_2 = 3 $: $ a_3 = 1 + 2(3) = 7 $.
Continue with $ n = 4 $, using $ a_3 = 7 $: $ a_4 = 1 + 2(7) = 15 $.
Finally, for $ n = 5 $, use $ a_4 = 15 $: $ a_5 = 1 + 2(15) = 31 $.

Each step uses the previous term to find the next, systematically unfolding the sequence term by term.

Problem 94

Problem 96

Other exercises in this chapter

Problem 93

The sequence $\mid a_{n}$ \\} is recursively defined. Compute $a_{n}$ for $n=1,2, \ldots, 5$ $$ a_{n+1}=-2 a_{n}, a_{0}=1 $$

View solution

Problem 94

The sequence $\mid a_{n}$ \\} is recursively defined. Compute $a_{n}$ for $n=1,2, \ldots, 5$ $$ a_{n+1}=-2 a_{n}, a_{0}=2 $$

View solution

Problem 96

The sequence $\mid a_{n}$ \\} is recursively defined. Compute $a_{n}$ for $n=1,2, \ldots, 5$ $$ a_{n+1}=4-2 a_{n}, a_{0}=\frac{4}{3} $$

View solution

Problem 97

The sequence $\mid a_{n}$ \\} is recursively defined. Compute $a_{n}$ for $n=1,2, \ldots, 5$ $$ a_{n+1}=\frac{a_{n}}{1+a_{n}}, a_{0}=1 $$

View solution