Charlene Kirkwood manages gift shop. She is single, earns
$1,243.49 each week, and claims
3 allowances.

 The correct options are

Part A : option (a) the common difference is -10.

Part B : option (c) 3,  -2,  -7,  -12,  .   .   .

Step-by-step explanation:  We are given to solve the following two problems

PART A :  

We are given to select the statement that best describes the relationship between the successive terms in the following sequence :

9,   -1,   -11,    -21,   .   .   .

If a(n) denotes the n-th term of the given sequence, then we see that

Since the difference between any two consecutive term of the given sequence is -10, so

there is a common difference of -10.

Option (a) is CORRECT.

PART B :  

Given that a sequence is defined recursively using th efollowing formula :

We are to select the sequence that could be generated using formula (i).

Let f(n) = 1. Substituting n = 1, 2, 3, 4,  .  .  . in equation (i), we get

In this case, the sequence is 1, -4, -9, -14,  .  .  .

Let f(n) = 2. Substituting n = 1, 2, 3, 4,  .  .  . in equation (i), we get

In this case, the sequence is 2, -3, -8, -13,  .   .   .

Let f(n) = 3. Substituting n = 1, 2, 3, 4,  .  .  . in equation (i), we get

In this case, the sequence is 3, -2, -7, -12,  .   .   .

Let f(n) = 4. Substituting n = 1, 2, 3, 4,  .  .  . in equation (i), we get

In this case, the sequence is 4,-1, -6, -11,  .   .   .

Thus, the correct sequence that could be generated is 3, -2, -7, -12,  .  .  .

Option (c) is CORRECT.