Question

40 rachel uses the following steps to solve for k in the equation $k^2 - 4k + 4 = 16$.
$k^2 - 4k + 4 = 16$
step 1: $(k - 2)^2 = 16$
step 2: $sqrt{(k - 2)^2} = sqrt{16}$
step 3: $k - 2 = 4$
step 4: $k - 2 + 2 = 4 + 2$
step 5: $k = 6$
which best describes rachel’s error?
a rachel’s error is in step 1; the left side of the equation is equal to $(k + 2)^2$.
b rachel’s error is in step 2; she should only have taken the square root of the left side of the equation.
c rachel’s error is in step 3; the right side of the equation is equal to +4 or -4.
d rachel’s error is in step 4; she should have subtracted 2 from both sides of the equation instead of adding.
41 the explicit equation of an arithmetic sequence is described below.
$a_n = -6n - 107$
which represents the domain of this arithmetic sequence?
a all rational numbers
b all integers
c all real numbers
d all natural numbers

Explanation:

Question 40

Step 1: Analyze Step 1

The left - hand side of the equation \(k^{2}-4k + 4\) is a perfect square trinomial. Using the formula \((a - b)^{2}=a^{2}-2ab + b^{2}\), where \(a = k\) and \(b = 2\), we have \((k - 2)^{2}=k^{2}-4k + 4\). So Step 1 is correct.

Step 2: Analyze Step 2

When solving an equation of the form \(x^{2}=y\) (here \(x=(k - 2)\) and \(y = 16\)), we take the square root of both sides. So \(\sqrt{(k - 2)^{2}}=\sqrt{16}\) is a valid operation. Step 2 is correct.

Step 3: Analyze Step 3

We know that if \(\sqrt{x^{2}}=\sqrt{y}\), then \(x=\pm\sqrt{y}\) (when \(y\geq0\)). Here, \(\sqrt{(k - 2)^{2}}=\vert k - 2\vert\) and \(\sqrt{16} = 4\). So \(\vert k - 2\vert=4\), which means \(k - 2=4\) or \(k - 2=-4\). Rachel only considered the positive case, so Step 3 is wrong.

Step 4: Analyze Step 4

If \(k - 2 = 4\), then adding 2 to both sides (\(k-2 + 2=4 + 2\)) is the correct way to solve for \(k\). Since the error is in Step 3, we don't need to analyze Step 4 for the error source.

An arithmetic sequence is a sequence of numbers in which the difference between consecutive terms is constant. The explicit formula for an arithmetic sequence is \(a_{n}=a_{1}+(n - 1)d\), where \(n\) represents the term number.

In the formula \(a_{n}=-6n-107\), the variable \(n\) represents the position of the term in the sequence. The domain of a sequence (the set of values that \(n\) can take) is the set of natural numbers (positive integers \(n = 1,2,3,\cdots\)) because we are talking about the 1st term, 2nd term, 3rd term, etc. of the sequence.

• Option A: Rational numbers include fractions and negative numbers. We can't have a negative or fractional term number in a sequence (we don't have a - 1st term or a \(\frac{1}{2}\)nd term), so this is incorrect.
• Option B: Integers include negative integers. We can't have a negative term number, so this is incorrect.
• Option C: Real numbers include all rational and irrational numbers, including non - positive and non - integer values. We can't have non - positive or non - integer term numbers, so this is incorrect.
• Option D: Natural numbers are positive integers (\(1,2,3,\cdots\)), which are the valid values for the term number \(n\) in a sequence.

Answer:

C. Rachel’s error is in step 3; the right side of the equation is equal to +4 or -4.

Question 41