Let's solve this problem step-by-step.

Problem:

We have an arithmetic sequence with the following details:

1. The sequence has 10 terms.
2. The sum of the first three terms is 45.
3. The sum of the last two terms is 105.
4. We need to find the total sum of all terms in this sequence.

Let:

• $a$ be the first term of the sequence.
• $d$ be the common difference.

Steps to Solve:

1. Express the first three terms' sum: $a + (a + d) + (a + 2d) = 45$ Simplifying, we get: $3a + 3d = 45$ a + d = 15 \tag{Equation 1}

2. Express the last two terms' sum: The 9th term is $a + 8d$ and the 10th term is $a + 9d$. $(a + 8d) + (a + 9d) = 105$ Simplifying, we get: 2a + 17d = 105 \tag{Equation 2}

3. Solve the system of equations (Equation 1 and Equation 2):

   From Equation 1: $a + d = 15 \Rightarrow a = 15 - d$

   Substitute $a = 15 - d$ into Equation 2: $2(15 - d) + 17d = 105$ $30 - 2d + 17d = 105$ $15d = 75$ $d = 5$

   Substitute $d = 5$ back into Equation 1: $a + 5 = 15$ $a = 10$

   So, $a = 10$ and $d = 5$.

4. Find the sum of the 10 terms:

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

   For $n = 10$: $S_{10} = \frac{10}{2} (2 \cdot 10 + (10 - 1) \cdot 5)$ $S_{10} = 5 (20 + 45)$ $S_{10} = 5 \cdot 65$ $S_{10} = 325$

Answer:

The total sum of all terms in the sequence is 325.

Would you like more details, or do you have any further questions?

Here are 5 related questions for further exploration:

1. What is the formula for the nth term of an arithmetic sequence?
2. How would you find the sum if only the first term and the common difference were given?
3. How do you derive the sum formula for an arithmetic sequence?
4. What if the sequence had more terms, say 20, and we knew a different sum of terms?
5. How can you check if a given sequence is arithmetic?

Tip: When dealing with arithmetic sequences, always look for relationships between terms to set up equations that can solve for unknowns.