The problem provided is about an arithmetic sequence, where two terms are given:

• $U_8 = 18$ (the 8th term is 18)
• $U_{15} = 46$ (the 15th term is 46)

The questions ask to determine:

1. The common difference ($d$) and the first term ($U_1$) of the sequence.
2. The 12th term of the sequence ($U_{12}$).
3. The sum of the first 10 terms of the sequence.

Solution:

For an arithmetic sequence, the general formula for the $n$-th term is: $U_n = U_1 + (n - 1) \cdot d$ Where:

• $U_n$ is the $n$-th term,
• $U_1$ is the first term,
• $d$ is the common difference,
• $n$ is the term's position.

Step 1: Finding the Common Difference ($d$) and the First Term ($U_1$)

We know the values of $U_8$ and $U_{15}$, so let's write two equations:

U_8 = U_1 + 7d = 18 \tag{1} U_{15} = U_1 + 14d = 46 \tag{2}

By subtracting equation (1) from equation (2): $(U_1 + 14d) - (U_1 + 7d) = 46 - 18$ $7d = 28$ $d = 4$

Now substitute $d = 4$ back into equation (1): $U_1 + 7 \cdot 4 = 18$ [ U