H.O.T. Problems Use Higher-Order Thinking Skills 32. OPEN ENDED Create an arithmetic sequence with a common difference of -10. 33. CCSS PERSEVERANCE Find the value of x that makes x + 8, 4x + 6, and 3x the first three terms of an arithmetic sequence. 34. REASONING Compare and contrast the domain and range of the linear functions described by Ax + By = C and a = a + (n – 1)d. 35. CHALLENGE Determine whether each sequence is an arithmetic sequence. Write yes or no. Explain. If yes, find the common difference and the next three terms. a. 2x + 1, 3x + 1, 4x + 1… b. 2x, 4x, 8x, …
The Correct Answer and Explanation is:
32. Create an arithmetic sequence with a common difference of -10.
Answer:
An arithmetic sequence is a sequence of numbers where the difference between consecutive terms is constant. This difference is called the common difference $d$.
If the common difference $d = -10$, an example arithmetic sequence could be:
$50, 40, 30, 20, 10, 0, -10, \dots$
Explanation:
- Starting from 50 (first term), each subsequent term is found by subtracting 10.
- $50 – 10 = 40$,
- $40 – 10 = 30$, and so forth.
Thus, this sequence fits the definition of an arithmetic sequence with a common difference of $-10$.
33. Find the value of $x$ that makes $x + 8$, $4x + 6$, and $3x$ the first three terms of an arithmetic sequence.
Answer:
For these three terms to form an arithmetic sequence, the difference between consecutive terms must be the same.
Set the common difference $d$ as:
$$
(4x + 6) – (x + 8) = 3x – (4x + 6)
$$
Simplify both sides:
$$
(4x + 6) – (x + 8) = 3x + 6 – 4x = -x + 6
$$
Calculate:
$$
(4x + 6) – (x + 8) = 3x + (6 – 8) = 3x – 2
$$
And
$$
3x – (4x + 6) = 3x – 4x – 6 = -x – 6
$$
Set equal:
$$
3x – 2 = -x – 6
$$
Add $x$ to both sides:
$$
3x + x – 2 = -6
$$
$$
4x – 2 = -6
$$
Add 2 to both sides:
$$
4x = -4
$$
Divide both sides by 4:
$$
x = -1
$$
34. Compare and contrast the domain and range of the linear functions described by $Ax + By = C$ and $a = a + (n – 1)d$.
Answer:
- Function $Ax + By = C$:
This is the standard form of a linear equation in two variables $x$ and $y$. The graph is a straight line. - Domain: All real values of $x$ for which $y$ can be found. Usually, $x \in \mathbb{R}$, unless restricted. So domain is typically all real numbers.
- Range: Similarly, since the line extends infinitely in both directions, $y$ can take all real values. So the range is also all real numbers.
- Function $a = a + (n – 1)d$:
This formula defines the nth term of an arithmetic sequence. - Domain: The input variable $n$ is the term number, so $n$ is restricted to positive integers (1, 2, 3, …), since terms in a sequence are counted discretely.
- Range: The values of $a$ (the terms of the sequence) depend on $a$ (first term) and $d$ (common difference). The range is the set of these term values and can be infinite or finite depending on $d$.
Contrast:
- The first is a continuous linear function with both domain and range as real numbers.
- The second is a discrete function (sequence), with domain as positive integers and range as the terms generated by the formula.
35. Determine whether each sequence is arithmetic. Write yes or no. Explain. If yes, find the common difference and the next three terms.
a. $2x + 1, 3x + 1, 4x + 1, \dots$
Calculate the difference between consecutive terms:
$$
(3x + 1) – (2x + 1) = 3x + 1 – 2x – 1 = x
$$
$$
(4x + 1) – (3x + 1) = 4x + 1 – 3x – 1 = x
$$
Since the difference between consecutive terms is the same ($x$), this is an arithmetic sequence with common difference $d = x$.
The next three terms would be:
$$
5x + 1, \quad 6x + 1, \quad 7x + 1
$$
b. $2x, 4x, 8x, \dots$
Calculate the difference between consecutive terms:
$$
4x – 2x = 2x
$$
$$
8x – 4x = 4x
$$
Since the difference is not constant (2x then 4x), this is not an arithmetic sequence.
Explanation:
An arithmetic sequence requires a constant difference between terms. This sequence doubles each time, indicating it’s geometric, not arithmetic.
Summary:
| Problem | Answer | Explanation Summary |
|---|---|---|
| 32 | Sequence example: 50, 40, 30… | Common difference -10 |
| 33 | $x = -1$ | Solve difference equality |
| 34 | Linear eqn: domain & range real; sequence: domain positive integers, range depends on terms | Continuous vs discrete function |
| 35a | Yes; common difference $x$; next terms $5x+1, 6x+1, 7x+1$ | Constant difference $x$ |
| 35b | No | Differences vary; not arithmetic |
