Question

write an exponential equation for the geometric sequence 1, 10, 100, 1000, ...
a) (a_n = 10(1)^{n - 1}
b) (a_n = 10(0.1)^{n - 1}
c) (a_n = 1(10)^{n - 1}
d) (a_n = 10(10)^{n - 1}
question 20 (5 points)
identify the first 4 terms in the arithmetic sequence given by the explicit formula (f(n)=8 + 3(n - 1)).
a) 3, -5, -13, -21
b) 3, 11, 19, 27
c) 8, 11, 14, 17

Explanation:

Step1: Recall geometric - sequence formula

The general formula for a geometric sequence is $a_{n}=a_{1}r^{n - 1}$, where $a_{1}$ is the first - term and $r$ is the common ratio.

Step2: Identify $a_{1}$ and $r$ for the given sequence

For the sequence $1,10,100,1000,\cdots$, $a_{1}=1$ and $r=\frac{10}{1}=10$.

Step3: Substitute values into the formula

Substitute $a_{1}=1$ and $r = 10$ into $a_{n}=a_{1}r^{n - 1}$, we get $a_{n}=1\times(10)^{n - 1}$.

for second question:

Step1: Recall arithmetic - sequence formula

The explicit formula for an arithmetic sequence is $f(n)=a_{1}+d(n - 1)$, where $a_{1}$ is the first - term and $d$ is the common difference. For $f(n)=8 + 3(n - 1)$, $a_{1}=8$ and $d = 3$.

Step2: Find the first term

When $n = 1$, $f(1)=8+3(1 - 1)=8$.

Step3: Find the second term

When $n = 2$, $f(2)=8+3(2 - 1)=8 + 3=11$.

Step4: Find the third term

When $n = 3$, $f(3)=8+3(3 - 1)=8+3\times2=8 + 6=14$.

Step5: Find the fourth term

When $n = 4$, $f(4)=8+3(4 - 1)=8+3\times3=8 + 9=17$.

Answer:

C. $a_{n}=1(10)^{n - 1}$