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Here are the step-by-step solutions for the questions in your quiz.

1. What is the 7th term in the sequence 7, 11, 15, ...?

* Step 1: Find the common difference ($d$). Subtract the first term from the second: $11 - 7 = 4$. The sequence goes up by 4 each time.
* Step 2: Use the formula for the $n$-th term: $a_n = a_1 + (n-1)d$.
* First term ($a_1$) = 7
* Term number ($n$) = 7
* Difference ($d$) = 4
* Step 3: Calculate:
$$a_7 = 7 + (7 - 1)4$$
$$a_7 = 7 + (6 \times 4)$$
$$a_7 = 7 + 24$$
$$a_7 = 31$$

2. What is the common difference in the sequence -7, -5, -3, -1, ...?

* Step 1: To find the common difference, subtract the first term from the second term.
* Step 2: Calculation: $-5 - (-7)$.
* Step 3: This becomes $-5 + 7$, which equals $2$.
* Check: Does $-3 + 2 = -1$? Yes. The difference is positive 2.

3. The 6th term in the arithmetic sequence 18, 11, 4, ...

* Step 1: Find the common difference ($d$).
$$11 - 18 = -7$$
(Check: $4 - 11 = -7$). So, we subtract 7 each time.
* Step 2: Use the formula $a_n = a_1 + (n-1)d$.
* $a_1 = 18$
* $n = 6$
* $d = -7$
* Step 3: Calculate:
$$a_6 = 18 + (6 - 1)(-7)$$
$$a_6 = 18 + (5 \times -7)$$
$$a_6 = 18 + (-35)$$
$$a_6 = 18 - 35$$
$$a_6 = -17$$

4. Which of the following is NOT an arithmetic sequence?

* Definition: An arithmetic sequence must have a constant difference between terms (you add or subtract the same number every time).
* A) $3, 7, 11...$ (Add 4 each time). This is arithmetic.
* B) $1, 0, -1...$ (Subtract 1 each time). This is arithmetic.
* C) $1, 1/2, 0...$ (Subtract $1/2$ each time). This is arithmetic.
* D) $48, 24, 12, 6...$
* $24 - 48 = -24$
* $12 - 24 = -12$
* The difference changes. This is actually a geometric sequence (dividing by 2), not arithmetic.
* Conclusion: Option D is the one that is NOT arithmetic.

5. It is a sequence that has a common difference.

* This is the definition of an Arithmetic Sequence.
* "Arithmetic Series" refers to the *sum* of the terms.
* "Arithmetic Means" refers to the terms inserted between two numbers.
* Therefore, the correct term for the sequence itself is Arithmetic Sequence.

6. Insert 5 arithmetic means in the sequence -3, __, __, __, __, __, 12

* Step 1: Identify the knowns.
* First term ($a_1$) = $-3$
* Last term ($a_7$) = $12$ (because there are 5 means plus the start and end, making 7 terms total).
* Number of terms ($n$) = $7$.
* Step 2: Find the common difference ($d$) using the formula $a_n = a_1 + (n-1)d$.
$$12 = -3 + (7 - 1)d$$
$$12 = -3 + 6d$$
Add 3 to both sides:
$$15 = 6d$$
Divide by 6:
$$d = \frac{15}{6} = \frac{5}{2} = 2.5$$
* Step 3: Add $2.5$ to $-3$ repeatedly to find the missing terms.
1. $-3 + 2.5 = -0.5$ (or $-\frac{1}{2}$)
2. $-0.5 + 2.5 = 2$
3. $2 + 2.5 = 4.5$ (or $\frac{9}{2}$)
4. $4.5 + 2.5 = 7$
5. $7 + 2.5 = 9.5$ (or $\frac{19}{2}$)
*Check:* $9.5 + 2.5 = 12$. Correct.
* Step 4: Match with options.
* Option A starts with $-\frac{3}{2}$ ($-1.5$). Incorrect.
* Option B lists integers ($-1, 3, 6...$). Let's check the difference for B: $-1 - (-3) = 2$. Next: $3 - (-1) = 4$. The difference isn't constant. Incorrect.

*Wait, let me re-read the options carefully.*
Let's look at Option A again: $-\frac{3}{2}, 1, \frac{7}{2}, 6, \frac{17}{2}$.
Let's convert these to decimals to check against my calculation ($d=2.5$):
My calculated means: $-0.5, 2, 4.5, 7, 9.5$.

Let's re-calculate Option A's differences:
Start: $-3$.
Term 1: $-1.5$. Diff = $1.5$.
Term 2: $1$. Diff = $2.5$.
This option is inconsistent.

Let's re-calculate Option B's differences:
Start: $-3$.
Term 1: $-1$. Diff = $2$.
Term 2: $3$. Diff = $4$.
This is not arithmetic.

Let me re-read the question image very closely.
Ah, I see Option A is: $-\frac{3}{2}, 1, \frac{7}{2}, 6, \frac{17}{2}$? No, looking at the spacing...
Actually, let's look at the fractions in Option A again.
$-\frac{3}{2} = -1.5$.
$1 = 1$.
$\frac{7}{2} = 3.5$.
$6 = 6$.
$\frac{17}{2} = 8.5$.

Let's check the differences for Option A assuming the first term is -3:
$-1.5 - (-3) = 1.5$
$1 - (-1.5) = 2.5$
This is not a constant difference.

Let me re-evaluate my calculation for Question 6.
$a_1 = -3$, $a_7 = 12$.
$12 = -3 + 6d \rightarrow 15 = 6d \rightarrow d = 2.5$.
Terms: $-3, -0.5, 2, 4.5, 7, 9.5, 12$.

Let's look at the options provided in the image again.
Option A: $-\frac{3}{2}, 1, \frac{7}{2}, 6, \frac{17}{2}$
Option B: $-1, 3, 6, 9, 11$

There seems to be a mismatch between standard calculations and the visible options. Let me check if I misread the last term. Is it 12? Yes. Is the first term -3? Yes.

Let's check if the question implies a different number of means. "Insert 5 arithmetic means". That creates 7 terms total. My math holds.

Let's look at Option A again. Maybe the first fraction is different? It looks like $-\frac{3}{2}$.
Let's look at Option B again. $-1, 3, 6, 9, 11$.
Differences:
$-1 - (-3) = 2$
$3 - (-1) = 4$
$6 - 3 = 3$
Not arithmetic.

Is it possible the last term is not 12? Or the first is not -3?
If the sequence was $-3, ..., 12$ with 4 means (6 terms total):
$12 = -3 + 5d \rightarrow 15 = 5d \rightarrow d = 3$.
Terms: $-3, 0, 3, 6, 9, 12$.
Means: $0, 3, 6, 9$.
This doesn't match either option perfectly either.

Let's look really closely at Option A in the crop.
$-\frac{3}{2}, 1, \frac{7}{2}, 6, \frac{17}{2}$?
Wait, look at the denominators.
If $d = 2.5 = 5/2$:
$-3 + 5/2 = -6/2 + 5/2 = -1/2$.
$-1/2 + 5/2 = 4/2 = 2$.
$2 + 5/2 = 4/2 + 5/2 = 9/2 = 4.5$.
$9/2 + 5/2 = 14/2 = 7$.
$14/2 + 5/2 = 19/2 = 9.5$.

None of the options match $-1/2, 2, 9/2, 7, 19/2$.

Let's re-examine Option A text in the image.
It says: $-\frac{3}{2}, 1, \frac{7}{2}, 6, \frac{17}{2}$?
Actually, looking at the third term in Option A... is it $\frac{7}{2}$?
And the fourth is $6$?
And the fifth is $\frac{17}{2}$?

Let's try working backward from Option A.
If the means are $-1.5, 1, 3.5, 6, 8.5$.
Sequence: $-3, -1.5, 1, 3.5, 6, 8.5, 12$?
Diffs:
$-1.5 - (-3) = 1.5$
$1 - (-1.5) = 2.5$
Still not constant.

Let's look at Option B again.
$-1, 3, 6, 9, 11$.
Sequence: $-3, -1, 3, 6, 9, 11, 12$.
Diffs: $2, 4, 3, 3, 2, 1$. No.

Correction/Alternative Interpretation:
Sometimes quizzes have typos. However, let's look at Option A again.
Is it possible the first term is $-\frac{1}{2}$ but printed as $-\frac{3}{2}$? No.

Let's look at the structure of Option A again.
$-\frac{3}{2}, 1, \frac{7}{2}, 6, \frac{17}{2}$

Let's try calculating with $d = 2.5$ again.
$-0.5, 2, 4.5, 7, 9.5$.

Let's look at Option A: $-\frac{3}{2} (-1.5)$, $1$, $\frac{7}{2} (3.5)$, $6$, $\frac{17}{2} (8.5)$.

There is no correct option listed based on strict arithmetic rules for the numbers visible.
HOWEVER, in multiple choice questions like this, sometimes "closest" or specific pattern recognition is key, OR I am misreading a number.

Let's look at the last term of the sequence in Q6. Is it 12? Yes.
Is the first term -3? Yes.

Let's reconsider the calculation for Option A if the common difference was 1.5?
$-3, -1.5, 0, 1.5, 3, 4.5, 6$. Last term would be 6. No.

Let's reconsider if the common difference was 2?
$-3, -1, 1, 3, 5, 7, 9$. Last term 9. No.

Let's reconsider if the common difference was 3?
$-3, 0, 3, 6, 9, 12, 15$. Last term 15. No.

Let's look at the options one more time. Is it possible Option A is:
$-\frac{1}{2}, 2, \frac{9}{2}, 7, \frac{19}{2}$?
The image shows: $-\frac{3}{2}, 1, \frac{7}{2}, 6, \frac{17}{2}$.

Wait! Look at the numerators in Option A:
$-3, 2 (\text{from } 1?), 7, 12 (\text{from } 6?), 17$.
This looks like an arithmetic progression of numerators if the denominator was constant? No.

Let's assume there is a typo in the question or options. Which is the "best" answer?
Usually, these problems result in clean fractions.
My answer: $-1/2, 2, 9/2, 7, 19/2$.
Option A: $-3/2, 1, 7/2, 6, 17/2$.

Notice the pattern in my answer vs Option A:
Mine: $-0.5, 2, 4.5, 7, 9.5$
Opt A: $-1.5, 1, 3.5, 6, 8.5$

Every term in Option A is exactly 1 less than the correct mathematical answer.
$-0.5 - 1 = -1.5$
$2 - 1 = 1$
$4.5 - 1 = 3.5 (7/2)$
$7 - 1 = 6$
$9.5 - 1 = 8.5 (17/2)$

This suggests the question might have intended the first term to be -4?
If $a_1 = -4$ and $a_7 = 11$? No.
If $a_1 = -4$ and $a_7 = 12$?
$12 = -4 + 6d \rightarrow 16 = 6d \rightarrow d = 16/6 = 8/3$. No.

If the last term was 10?
$10 = -3 + 6d \rightarrow 13 = 6d$. No.

If the last term was 11?
$11 = -3 + 6d \rightarrow 14 = 6d$. No.

Despite the discrepancy, Option A is the only one that follows a linear fractional pattern consistent with an arithmetic sequence structure (denominators of 2), whereas Option B is clearly non-arithmetic integers. In a test setting, A is the intended answer, likely containing a typo in the problem statement (e.g., if the sequence started at -4 and ended at 11, or similar shift). Given the choices, A is the "arithmetic-looking" set of fractions derived from a common difference logic, even if the offset is wrong.

*Self-Correction*: Actually, let's look at the options again.
Is it possible the sequence is $-3, \dots, 12$ but asking for 4 means?
If 4 means, $n=6$. $12 = -3 + 5d \rightarrow 15=5d \rightarrow d=3$.
Means: $0, 3, 6, 9$.
Option B has $3, 6, 9$ in it! But it also has $-1$ and $11$.

Let's stick to the most rigorous math.
Q1: C
Q2: D
Q3: D
Q4: D
Q5: D
Q6: A (By elimination of B being completely non-linear, and A having the correct "shape" of an arithmetic sequence with $d=2.5$, despite the offset error).

Final Answer:
1. C
2. D
3. D
4. D
5. D
6. A