Question

which of the following is an arithmetic sequence where a1 = 9 and d = 7? a) 0, 9, 18, 27, 36, ... b) 0, 7, 14, 21, 28, ... c) 9, 16, 23, 30, 37, ... d) 7, 16, 25, 34, 43, ... question 14 (5 points) identify the recursive formula for the sequence -3, 9, -27, 81, ... a) f(n) = {f(1)= -3; f(n)= -f(n - 1) if n>1} b) f(n) = {f(1)= 3; f(n)= -3f(n - 1) if n>1} c) f(n) = {f(1)= -3; f(n)= -3f(n - 1) if n>1}

Explanation:

Step1: Recall arithmetic - sequence formula

The general form of an arithmetic sequence is $a_n=a_1+(n - 1)d$, where $a_1$ is the first - term and $d$ is the common difference. Given $a_1 = 9$ and $d = 7$.

Step2: Check each option's first - term

For option A, the first - term is $0
eq9$. For option B, the first - term is $0
eq9$. For option C, the first - term is $9$. For option D, the first - term is $7
eq9$.

Step3: Check the common difference in option C

In option C, $16−9 = 7$, $23−16 = 7$, $30−23 = 7$, $37−30 = 7$. The common difference is $7$.

Step4: Recall recursive formula for geometric sequence

For a geometric sequence $a_n$ with first - term $a_1$ and common ratio $r$, the recursive formula is $f(1)=a_1$ and $f(n)=r\times f(n - 1)$ for $n>1$. For the sequence $-3,9, - 27,81,\cdots$, $a_1=-3$ and $r=\frac{9}{-3}=-3$.

Step5: Determine the recursive formula

The recursive formula is $f(1)=-3$ and $f(n)=-3\times f(n - 1)$ for $n > 1$.

Answer:

C. 9, 16, 23, 30, 37,...
C. $f(n)=

$$\begin{cases}f(1)=-3\\f(n)=-3f(n - 1)\text{ if }n>1\end{cases}$$

$