Question

1. what is the seventh term of the sequence -2, 6, -18, 54,...? (1) -1458 (2) -4374 (3) 1458 (4) 4374 14. given x ≠ 0, where m(x) = 12x^(8a) and p(x) = 3x^(2a), the expression m(x)/p(x) is equivalent to (1) 9x^(4a) (2) 4x^(6a) (3) 4x^6 (4) 4x^4 15. which statement best describes the end behavior of the function y = log(x - 3)? (1) as x → -∞, y → -∞, and as x → ∞, y → ∞. (2) as x → 3, y → -∞, and as x → ∞, y → ∞. (3) as x → -∞, y → 0, and as x → ∞, y → ∞. (4) as x → 3, y → 0, and as x → ∞, y → ∞.

Explanation:

Step1: Identify sequence type (geometric)

The sequence $-2, 6, -18, 54,...$ is geometric with first term $a_1=-2$, common ratio $r=\frac{6}{-2}=-3$.

Step2: Use geometric term formula

The $n$-th term formula is $a_n=a_1r^{n-1}$. For $n=7$:
$a_7=(-2)\times(-3)^{7-1}=(-2)\times(-3)^6$

Step3: Calculate the power and product

$(-3)^6=729$, so $a_7=(-2)\times729=-1458$

---

Step1: Substitute functions into the ratio

$\frac{m(x)}{p(x)}=\frac{12x^{8a}}{3x^{2a}}$

Step2: Divide coefficients and variables

Divide coefficients: $\frac{12}{3}=4$. Use exponent rule $\frac{x^m}{x^n}=x^{m-n}$: $x^{8a-2a}=x^{6a}$. Combine: $4x^{6a}$

---

Step1: Identify log function domain

For $y=\log(x-3)$, the domain is $x>3$ (since argument $x-3>0$).

Step2: Analyze end behavior

As $x$ approaches 3 from the right ($x
ightarrow3^+$), $x-3
ightarrow0^+$, so $\log(x-3)
ightarrow-\infty$. As $x
ightarrow\infty$, $x-3
ightarrow\infty$, so $\log(x-3)
ightarrow\infty$.

Answer:

1. (1) $-1458$
2. (2) $4x^{6a}$
3. (2) As $x

ightarrow 3, y
ightarrow -\infty$, and as $x
ightarrow \infty, y
ightarrow \infty$.