Let's work through the questions from the uploaded image.

Arithmetic Series

The general formula for the nth term $a_n$ and sum $S_n$ of an arithmetic series are:

• $a_n = a_1 + (n-1)d$
• $S_n = \frac{n}{2} \cdot (a_1 + a_n)$

Where:

• $a_1$ is the first term
• $d$ is the common difference
• $n$ is the number of terms

1. $7, 11, 15, \dots$

• Common difference $d = 11 - 7 = 4$

• Find $a_7$:
  $a_7 = 7 + (7-1) \times 4 = 7 + 24 = 31$

• Find $S_7$:
  $S_7 = \frac{7}{2} \times (7 + 31) = \frac{7}{2} \times 38 = 133$

2. $18, 11, 4, \dots$

• Common difference $d = 11 - 18 = -7$

• Find $a_6$:
  $a_6 = 18 + (6-1) \times (-7) = 18 - 35 = -17$

• Find $S_6$:
  $S_6 = \frac{6}{2} \times (18 + (-17)) = 3 \times 1 = 3$

3. $-7, -5, -3, \dots$

• Common difference $d = -5 - (-7) = 2$

• Find $a_{23}$:
  $a_{23} = -7 + (23-1) \times 2 = -7 + 44 = 37$

• Find $S_{23}$:
  $S_{23} = \frac{23}{2} \times (-7 + 37) = \frac{23}{2} \times 30 = 345$

4. $3, 3 \frac{1}{2}, 4, \dots$

• Common difference $d = 3.5 - 3 = 0.5$

• Find $a_{16}$:
  $a_{16} = 3 + (16-1) \times 0.5 = 3 + 7.5 = 10.5$

• Find $S_{16}$:
  $S_{16} = \frac{16}{2} \times (3 + 10.5) = 8 \times 13.5 = 108$

5. $-17, -12, -7, \dots$

• Common difference $d = -12 - (-17) = 5$

• Find $a_{24}$:
  $a_{24} = -17 + (24-1) \times 5 = -17 + 115 = 98$

• Find $S_{24}$:
  $S_{24} = \frac{24}{2} \times (-17 + 98) = 12 \times 81 = 972$

Geometric Series

The general formula for the nth term $a_n$ and sum $S_n$ of a geometric series are:

• $a_n = a_1 \times r^{n-1}$
• $S_n = a_1 \times \frac{1 - r^n}{1 - r}$ (for $|r| < 1$)
• Sum of infinite series $S_{\infty} = \frac{a_1}{1 - r}$ (for $|r| < 1$)

1. $-1, 6, -36, 216, \dots$

• Common ratio $r = \frac{6}{-1} = -6$

• Find $a_{19}$:
  $a_{19} = -1 \times (-6)^{18} = -6^{18}$ $a_{19}$ is a very large value.

• Find $S_{19}$ using the sum formula for a geometric series (not suitable for this large common ratio).

2. $\frac{1}{2}, \frac{1}{8}, \frac{1}{32}, \dots$

• Common ratio $r = \frac{1/8}{1/2} = \frac{1}{4}$

• Find $a_{27}$:
  $a_{27} = \frac{1}{2} \times \left(\frac{1}{4}\right)^{26}$

• Find $S_{27}$ and infinite sum (since $|r| < 1$) using geometric formulas.

Would you like detailed calculations for the geometric series or any other part?

Here are 5 related questions:

1. What is the nth term formula for a quadratic sequence?
2. Can the sum formula for an arithmetic series apply to negative sequences?
3. What happens if the common difference in an arithmetic sequence is fractional?
4. How do you determine the common ratio for fractional geometric sequences?
5. What is the application of infinite series sums in real-life scenarios?

Tip: In geometric series, ensure the common ratio's magnitude is less than 1 to use the infinite sum formula effectively.