Question

29 the recursive formula of an arithmetic sequence is described below.
$a_1 = 19$, $a_n = a_{n - 1} + 6$
which represents the explicit equation of this arithmetic sequence?
a $a_n = 6n + 13$
b $a_n = 6n + 19$
c $a_n = 13n + 6$
d $a_n = 19n + 6$

30 given the function $q(x) = ax^2 + bx + c$, and $q(-5) = 0$, which of the following must be true?
a the binomial $(x + 5)$ is a factor of $ax^2 + bx + c$.
b the binomial $(x - 5)$ is a factor of $ax^2 + bx + c$.
c the number 5 is a zero of the function $q(x)$.
d the graph of $q(x)$ has a maximum at $x = -5$.

31 to teach miguel about saving money, his father told him that he would hold some of miguel’s money and double the amount each year, $x$, but he would only hold it so long as the amount was still under $100. if miguel gives his father $1 and his father doubles the money each year, so long as the doubled amount is below $100, which inequality could miguel use to track the amount of money?
a $1 \cdot x^2 < 100$
b $1 \cdot x^2 > 100$
c $1 \cdot 2^x < 100$
d $1 \cdot 2^x > 100$

Explanation:

Question 29

Step1: Recall arithmetic sequence formula

The explicit formula for an arithmetic sequence is \(a_n = a_1 + (n - 1)d\), where \(a_1\) is the first term and \(d\) is the common difference.
Here, \(a_1 = 19\) and \(d = 6\) (from \(a_n=a_{n - 1}+6\)).

Step2: Substitute values into formula

Substitute \(a_1 = 19\) and \(d = 6\) into \(a_n=a_1+(n - 1)d\):

$$
LATEXBLOCK0
$$

Step1: Recall factor theorem

The factor theorem states that if \(q(c)=0\), then \((x - c)\) is a factor of \(q(x)\). Here, \(q(-5)=0\), so \(c=-5\).

Step2: Apply factor theorem

Using the factor theorem, if \(q(-5) = 0\), then \((x-(-5))=(x + 5)\) is a factor of \(q(x)=ax^2+bx + c\).

• Option B: \((x - 5)\) would be a factor if \(q(5)=0\), not \(q(-5)=0\), so B is wrong.
• Option C: A zero at \(x = 5\) would mean \(q(5)=0\), not \(q(-5)=0\), so C is wrong.
• Option D: We don't know if the parabola opens up or down, so we can't say if \(x=-5\) is a maximum. So D is wrong.

Step1: Analyze the growth model

Miguel starts with \(\$1\) and the amount doubles each year. This is an exponential growth model. The formula for exponential growth is \(A = P\times r^x\), where \(P\) is the principal amount, \(r\) is the growth rate, and \(x\) is the time.
Here, \(P = 1\), \(r = 2\) (since it doubles each year), so the amount after \(x\) years is \(1\times2^x\).

Step2: Set up the inequality

The amount must be under \(\$100\), so \(1\times2^x<100\).

Answer:

A. \(a_n = 6n + 13\)

Question 30