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Chapter 19: Problem 3

A sequence is defined by $$ x[k]=\frac{k^{2}}{2}+k, \quad k=0,1,2,3, \ldots $$ State the first five terms.

Short Answer

Expert verified
The first five terms are 0, \(\frac{3}{2}\), 4, \(\frac{15}{2}\), 12.

Step by step solution

01

Understand the sequence formula

The sequence is defined by the expression \(x[k] = \frac{k^2}{2} + k\), where \(k\) is a non-negative integer (0, 1, 2, 3, ...). Our task is to find the first five terms.
02

Calculate the term for \(k = 0\)

Substitute \(k = 0\) into the formula:\[ x[0] = \frac{0^2}{2} + 0 = 0. \] So, the first term is 0.
03

Calculate the term for \(k = 1\)

Substitute \(k = 1\) into the formula:\[ x[1] = \frac{1^2}{2} + 1 = \frac{1}{2} + 1 = \frac{3}{2}. \] Thus, the second term is \(\frac{3}{2}\).
04

Calculate the term for \(k = 2\)

Substitute \(k = 2\) into the formula:\[ x[2] = \frac{2^2}{2} + 2 = 2 + 2 = 4. \] The third term is 4.
05

Calculate the term for \(k = 3\)

Substitute \(k = 3\) into the formula:\[ x[3] = \frac{3^2}{2} + 3 = \frac{9}{2} + 3 = \frac{9}{2} + \frac{6}{2} = \frac{15}{2}. \] Hence, the fourth term is \(\frac{15}{2}\).
06

Calculate the term for \(k = 4\)

Substitute \(k = 4\) into the formula:\[ x[4] = \frac{4^2}{2} + 4 = \frac{16}{2} + 4 = 8 + 4 = 12. \] The fifth term is 12.

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Key Concepts

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

Sequence Formula
A sequence formula is a mathematical expression that helps you determine the terms of a sequence. In our exercise, the sequence is defined by the formula \(x[k] = \frac{k^2}{2} + k\). This formula tells us how to generate each term based on the variable \(k\), which represents a position in the sequence.
  • The beauty of using a sequence formula is its ability to extend a sequence as far as you need, by substituting different values for \(k\).
  • Here, \(k\) can be any non-negative integer: 0, 1, 2, 3, and so on.
Understanding the rules provided by the formula is key to finding specific sequence terms. If you grasp how the formula works, you can easily generate the desired terms.
Sequence Terms
Sequence terms are individual elements found at certain positions in a sequence. Each term is derived by plugging in a specific value of \(k\) into the sequence formula. In our case, to find the first five terms, we calculate as follows:
  • When \(k = 0\), the term is \(x[0] = 0\).
  • For \(k = 1\), the term is \(x[1] = \frac{3}{2}\).
  • With \(k = 2\), the term is \(x[2] = 4\).
  • At \(k = 3\), the term becomes \(x[3] = \frac{15}{2}\).
  • Finally, when \(k = 4\), the term is \(x[4] = 12\).
By finding these terms, you unlock a part of the sequence that is crucial for many mathematical problems, providing insight into patterns and trends over successive terms.
Arithmetic Sequence
An arithmetic sequence is a special type of sequence where each consecutive term increases or decreases by a constant value. In contrast, the sequence in our example is not strictly arithmetic because the difference between successive terms is not constant.
While a typical arithmetic sequence follows the format \(a, a+d, a+2d, \ldots\), where \(d\) is the common difference, our sequence deviates from this pattern.
  • The sequence begins with terms: 0, \(\frac{3}{2}\), 4, \(\frac{15}{2}\), and 12.
  • Each difference between terms varies: from 0 to \(\frac{3}{2}\) is \(\frac{3}{2}\), from \(\frac{3}{2}\) to 4 is \(\frac{5}{2}\), and so on.
Even though not an arithmetic sequence, understanding this distinction aids in comprehending broader mathematical concepts. Not all sequences fit into perfect linear patterns, emphasizing the importance of considering various sequence types in mathematics.

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