Question

1. given the following pattern: 3, 7, 11, 15, 19, ...

a. identify whether the sequence is arithmetic or geometric. (1 point)
b. find the common difference, d, or the common ratio, r. (1 point)
c. write the explicit formula for the pattern above. (1 point)

1. given the following pattern 15, 45, 135, 405, 1215, ...

a. identify whether the sequence is arithmetic or geometric. (1 point)
b. find the common difference, d, or the common ratio, r. (1 point)
c. write the explicit formula for the pattern above. (1 point)

1. a line passes through the point (8, 12) and has a slope of 3. write the equation in point - slope form of this line. (2 points)

Explanation:

1. For the sequence 3, 7, 11, 15, 19, ...

Step1: Identify the sequence type

Check the difference between consecutive terms. $7 - 3=4$, $11 - 7 = 4$, $15 - 11=4$, $19 - 15 = 4$. Since the difference is constant, it is an arithmetic sequence.

Step2: Find the common - difference

The common difference $d$ is found by subtracting consecutive terms. As calculated above, $d = 4$.

Step3: Write the explicit 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, $a_1 = 3$ and $d = 4$, so $a_n=3+(n - 1)\times4=3 + 4n-4=4n - 1$.

2. For the sequence 15, 45, 135, 405, 1215, ...

Step1: Identify the sequence type

Check the ratio between consecutive terms. $\frac{45}{15}=3$, $\frac{135}{45}=3$, $\frac{405}{135}=3$, $\frac{1215}{405}=3$. Since the ratio is constant, it is a geometric sequence.

Step2: Find the common ratio

The common ratio $r$ is found by dividing consecutive terms. As calculated above, $r = 3$.

Step3: Write the explicit formula

The explicit formula for a geometric sequence is $a_n=a_1r^{n - 1}$, where $a_1$ is the first term and $r$ is the common ratio. Here, $a_1 = 15$ and $r = 3$, so $a_n=15\times3^{n - 1}$.

3. For the line passing through (8, 12) with slope 3

Step1: Recall the point - slope form

The point - slope form of a line is $y - y_1=m(x - x_1)$, where $(x_1,y_1)$ is a point on the line and $m$ is the slope.

Step2: Substitute the values

Substitute $x_1 = 8$, $y_1 = 12$, and $m = 3$ into the point - slope form. We get $y - 12=3(x - 8)$.

Answer:

1.
a. Arithmetic sequence
b. $d = 4$
c. $a_n=4n - 1$
2.
a. Geometric sequence
b. $r = 3$
c. $a_n=15\times3^{n - 1}$

1. $y - 12=3(x - 8)$