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Chapter 3: Problem 132

For the sequence z defined by $$z_{n}=(2+n) 3^{n}, \quad n \geq 0$$. Find \(z_{3}\)

Short Answer

Expert verified
The value of \(z_{3}\) is 135

Step by step solution

01

Understanding the sequence rule

The sequence rule given is \(z_{n}=(2+n) 3^{n}\). It tells us how to find any term in the sequence. \(n\) represents the index of the term. For example, if \(n = 0\), we're looking at the first term in the sequence. If \(n = 3\), we're looking at the fourth term.
02

Substitute n with the required term index

The task is to find \(z_{3}\), which means we need to substitute \(n\) with 3 in the sequence rule. So we get \(z_{3}=(2+3) 3^{3}\).
03

Calculate the result

Now, we can calculate the result: \(z_{3}=(2+3) 3^{3}\) equals \(z_{3}=5 \times 27\), which equals \(z_{3}=135\).

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Key Concepts

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

Geometric Sequences
Geometric sequences are a type of mathematical sequence where each term is found by multiplying the previous term by a fixed number, known as the "common ratio." A classic example is the sequence where each term is twice the preceding one, like 2, 4, 8, 16. Here, the common ratio is 2.
This pattern of multiplying by a constant applies to many real-world scenarios, such as exponential growth in populations or calculating compound interest.
  • Each term, except the first, depends on the former by multiplying by a constant.
  • Geometric sequences often involve powers of a fixed number, differing from arithmetic sequences, where terms have a constant difference.
Understanding geometric sequences is crucial in identifying patterns and predicting future terms. It aids in recognizing whether a number sequence follows a geometric progression or is driven by another mathematical rule.
Index of Sequence
The index of a sequence is simply the position of a term within a sequence. The index is usually denoted by the letter \(n\) and helps to find any specific term in the sequence. For instance, the index 0 would refer to the first term, \(n = 1\) to the second term, and so on.
In our particular sequence given as \(z_{n}=(2+n) 3^{n}\), \(n\) determines the sequence position as well as varying the formula's components.
  • A sequential number, called the index, defines the term's order and is essential for calculations within the sequence rule.
  • Changing the index alters the term value, calculated based on the sequence formula.
Therefore, understanding how to interpret and apply the index is key to working with sequences and deriving any term needed.
Sequence Rule
A sequence rule is a mathematical expression or formula that defines the progression of a sequence. It provides a way to calculate any specific term based on its index.
In the case of the sequence given by \(z_{n}=(2+n) 3^{n}\), this is the sequence rule that determines each term's value.
  • The sequence rule combines both the position \(n\) and mathematical operations (addition and exponentiation) to create a term.
  • It is like a recipe, where following the instructions allows you to pinpoint the exact value of any term.
By substituting \(n\) into the sequence rule, you can derive any term in the sequence. For instance, for \(n = 3\), you get \(z_{3}=(2+3) 3^{3}\), which calculates as 135 after executing the arithmetic operations.

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Most popular questions from this chapter

For the sequence a defined by \(a_{n}=\frac{n-1}{n^{2}(n-2)^{2}}, \quad n \geq 3\) and the sequence \(z\) defined by \(z_{n}=\sum_{i=3}^{n} a_{i}\). Is \(a\) decreasing?

In if the statement is true for all real numbers, prove it; otherwise, give a counterexample. $$ \lceil x+3\rceil=\lceil x\rceil+3 $$

For the sequence \(r\) defined by $$r_{n}=3 \cdot 2^{n}-4 \cdot 5^{n}, \quad n \geq 0$$. Find \(r_{0}\).

Prove that if \(n\) is an odd integer, $$ \left.\mid \frac{n^{2}}{4}\right\rfloor=\left(\frac{n-1}{2}\right)\left(\frac{n+1}{2}\right) $$

List all strings over \(X=\\{0,1\\}\) of length 2 or less.
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