To find the sum of an arithmetic sequence, we use the formula for the sum of an arithmetic series:

$S_n = \frac{n}{2} \times (a_1 + a_n)$

Where:

• $S_n$ is the sum of the first $n$ terms.
• $a_1$ is the first term.
• $a_n$ is the $n$-th term.
• $n$ is the number of terms.

Alternatively, we can also use the general formula for the $n$-th term of an arithmetic sequence:

$a_n = a_1 + (n-1)d$

Where:

• $a_n$ is the $n$-th term.
• $a_1$ is the first term.
• $d$ is the common difference.
• $n$ is the term number.

Let’s solve each sequence one by one.

1. Sequence: $3, 8 \frac{1}{2}, ..., A_{20}$

• First term: $a_1 = 3$
• Second term: $a_2 = 8 \frac{1}{2} = 8.5$
• Common difference: $d = 8.5 - 3 = 5.5$

Now, find the 20th term $a_{20}$: $a_{20} = 3 + (20 - 1) \times 5.5 = 3 + 19 \times 5.5 = 3 + 104.5 = 107.5$

Now, calculate the sum $S_{20}$: $S_{20} = \frac{20}{2} \times (3 + 107.5) = 10 \times 110.5 = 1105$

2. Sequence: $19, 13, ..., A_{10}$

• First term: $a_1 = 19$
• Second term: $a_2 = 13$
• Common difference: $d = 13 - 19 = -6$

Find the 10th term $a_{10}$: $a_{10} = 19 + (10 - 1) \times (-6) = 19 - 54 = -35$

Now, calculate the sum $S_{10}$: $S_{10} = \frac{10}{2} \times (19 + (-35)) = 5 \times (-16) = -80$

3. Sequence: $-17, -12, -7, ..., 33$

• First term: $a_1 = -17$
• Common difference: $d = -12 - (-17) = 5$
• Last term: $a_n = 33$

To find the number of terms $n$, use the formula for the $n$-th term: $33 = -17 + (n-1) \times 5$ $33 + 17 = (n-1) \times 5$ $50 = (n-1) \times 5 \quad \Rightarrow \quad n-1 = 10 \quad \Rightarrow \quad n = 11$

Now, calculate the sum $S_{11}$: $S_{11} = \frac{11}{2} \times (-17 + 33) = \frac{11}{2} \times 16 = 88$

4. Sequence: $9, -1, ..., A_8$

• First term: $a_1 = 9$
• Second term: $a_2 = -1$
• Common difference: $d = -1 - 9 = -10$

Find the 8th term $a_8$: $a_8 = 9 + (8 - 1) \times (-10) = 9 - 70 = -61$

Now, calculate the sum $S_8$: $S_8 = \frac{8}{2} \times (9 + (-61)) = 4 \times (-52) = -208$

5. Sequence: $4, 11, ..., A_{16}$

• First term: $a_1 = 4$
• Second term: $a_2 = 11$
• Common difference: $d = 11 - 4 = 7$

Find the 16th term $a_{16}$: $a_{16} = 4 + (16 - 1) \times 7 = 4 + 105 = 109$

Now, calculate the sum $S_{16}$: $S_{16} = \frac{16}{2} \times (4 + 109) = 8 \times 113 = 904$

Final Results:

1. $S_{20} = 1105$
2. $S_{10} = -80$
3. $S_{11} = 88$
4. $S_8 = -208$
5. $S_{16} = 904$

Would you like any further details on the steps or concepts used?

Here are five related questions:

1. How do you derive the common difference for an arithmetic sequence?
2. What is the formula to find the sum of the first $n$ terms of an arithmetic sequence?
3. How do you determine the number of terms in a sequence if you know the first and last terms?
4. What is the relationship between the common difference and the progression of an arithmetic sequence?
5. How can you calculate the sum of an arithmetic sequence with a negative common difference?

Tip: For arithmetic sequences, once you know the common difference, calculating any term is straightforward using the general formula.