To solve this problem, let's go through it step-by-step.

Given information:

1. The sequence is an arithmetic sequence with 10 terms.
2. The sum of the first three terms is 45.
3. The sum of the last two terms is 105.

We need to find the sum of all terms in this sequence.

Step 1: Define the Terms

Let's define the first term as $a$ and the common difference as $d$.

In an arithmetic sequence, the $n$-th term $T_n$ is given by: $T_n = a + (n - 1) \cdot d$

So, the first few terms are:

• First term: $T_1 = a$
• Second term: $T_2 = a + d$
• Third term: $T_3 = a + 2d$
• Ninth term: $T_9 = a + 8d$
• Tenth term: $T_{10} = a + 9d$

Step 2: Use the Sum of the First Three Terms

The sum of the first three terms is: $T_1 + T_2 + T_3 = a + (a + d) + (a + 2d) = 3a + 3d$ We know that this sum is 45, so: $3a + 3d = 45$ Dividing by 3: $a + d = 15$

Step 3: Use the Sum of the Last Two Terms

The sum of the last two terms (the 9th and 10th terms) is: $T_9 + T_{10} = (a + 8d) + (a + 9d) = 2a + 17d$ We know this sum is 105, so: $2a + 17d = 105$

Step 4: Solve the System of Equations

We now have two equations:

1. $a + d = 15$
2. $2a + 17d = 105$

From the first equation, solve for $a$: $a = 15 - d$ Substitute $a = 15 - d$ into the second equation: $2(15 - d) + 17d = 105$ $30 - 2d + 17d = 105$ $30 + 15d = 105$ $15d = 75$ $d = 5$ Substitute $d = 5$ back into $a + d = 15$: $a + 5 = 15$ $a = 10$

Step 5: Calculate the Sum of All Terms

The sum $S_n$ of the first $n$ terms in an arithmetic sequence is given by: $S_n = \frac{n}{2} \cdot (2a + (n - 1) \cdot d)$ For $n = 10$, $a = 10$, and $d = 5$: $S_{10} = \frac{10}{2} \cdot (2 \cdot 10 + (10 - 1) \cdot 5)$ $S_{10} = 5 \cdot (20 + 45)$ $S_{10} = 5 \cdot 65$ $S_{10} = 325$

Answer

The sum of all terms in the sequence is 325.

Would you like more details or have any questions?

Here are some related questions for further understanding:

1. How can we verify the solution by calculating each term individually?
2. What would happen if the sequence had a different number of terms?
3. How does the formula for the sum of an arithmetic sequence change if the first term and common difference are different?
4. What are other methods for solving systems of linear equations in arithmetic problems?
5. How does this solution change if the sequence is geometric instead of arithmetic?

Tip: In any arithmetic sequence problem, try setting up equations for known sums to solve for the first term and common difference easily.