Let's break down the problem and solve it step by step.

Part 1: Sum of the first 12 terms of the arithmetic sequence where $a_n = 3n + 5$

Step 1: Identify the first term $a_1$ and the common difference $d$

The general form of the sequence is given by: $a_n = 3n + 5$

For $n = 1$: $a_1 = 3(1) + 5 = 8$

For $n = 2$: $a_2 = 3(2) + 5 = 11$

The common difference $d$ is: $d = a_2 - a_1 = 11 - 8 = 3$

Step 2: Use the sum formula for the first 12 terms $S_{12}$

The sum of the first $n$ terms of an arithmetic sequence is given by: $S_n = \frac{n}{2} \times (2a_1 + (n - 1)d)$

For $n = 12$: $S_{12} = \frac{12}{2} \times (2(8) + (12 - 1) \times 3)$ $S_{12} = 6 \times (16 + 33) = 6 \times 49 = 294$

So, the sum of the first 12 terms is 294.

Part 2: Find the sum of 10 terms where the 3rd term is 12 and the 7th term is 8

Given:

• The 3rd term $a_3 = 12$
• The 7th term $a_7 = 8$

Step 1: Formulate the equations

For an arithmetic sequence, the general term is given by: $a_n = a_1 + (n - 1)d$

So, we can write the following equations: $a_3 = a_1 + 2d = 12 \quad \text{(Equation 1)}$ $a_7 = a_1 + 6d = 8 \quad \text{(Equation 2)}$

Step 2: Solve the system of equations

Subtract Equation 1 from Equation 2: $(a_1 + 6d) - (a_1 + 2d) = 8 - 12$ $4d = -4$ $d = -1$

Now, substitute $d = -1$ into Equation 1: $a_1 + 2(-1) = 12$ $a_1 - 2 = 12 \quad \Rightarrow \quad a_1 = 14$

So, the first term $a_1 = 14$ and the common difference $d = -1$.

Step 3: Find the sum of the first 10 terms

Use the sum formula again: $S_{10} = \frac{10}{2} \times (2a_1 + (10 - 1)d)$ $S_{10} = 5 \times (2(14) + 9(-1)) = 5 \times (28 - 9) = 5 \times 19 = 95$

So, the sum of the first 10 terms is 95.

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Would you like to see more details on any part of this solution? Here are some related questions to consider:

1. What is the sum of the first 15 terms of the sequence $a_n = 3n + 5$?
2. How does the sum change if the common difference is doubled?
3. How do we derive the sum formula for an arithmetic sequence?
4. What is the sum of the first 20 terms if the 4th term is 13 and the 8th term is 5?
5. How would the problem change if it were a geometric sequence instead?

Tip: Always verify the first term and common difference before applying the sum formula to avoid mistakes.