This problem involves finding the first term and the common difference of an arithmetic sequence.

We are given two pieces of information:

• The fourth term of the sequence is 17.
• The tenth term of the sequence is 47.

In an arithmetic sequence, the $n$-th term is given by the formula:

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

Where:

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

Step 1: Write equations for the 4th and 10th terms.

For the 4th term: $a_4 = a_1 + (4-1) \cdot d = 17$ $a_1 + 3d = 17 \quad \text{(Equation 1)}$

For the 10th term: $a_{10} = a_1 + (10-1) \cdot d = 47$ $a_1 + 9d = 47 \quad \text{(Equation 2)}$

Step 2: Solve the system of equations.

Now, we have the system of two linear equations:

$a_1 + 3d = 17 \quad \text{(1)}$ $a_1 + 9d = 47 \quad \text{(2)}$

By subtracting Equation 1 from Equation 2, we can eliminate $a_1$ and solve for $d$:

$(a_1 + 9d) - (a_1 + 3d) = 47 - 17$ $6d = 30$ $d = 5$

Step 3: Substitute $d = 5$ into Equation 1 to find $a_1$:

$a_1 + 3 \cdot 5 = 17$ $a_1 + 15 = 17$ $a_1 = 2$

Final Answer:

• The first term $a_1$ is 2.
• The common difference $d$ is 5.

Would you like a more detailed explanation or further assistance?

Here are 5 related questions for further practice:

1. How would you find the sum of the first 20 terms of this arithmetic sequence?
2. If the 15th term of the sequence is given, how would you calculate it?
3. How can the common difference of an arithmetic sequence affect the rate of growth?
4. What if the 7th term of an arithmetic sequence is given instead of the 4th—how would you approach it?
5. How would you graph the terms of this arithmetic sequence?

Tip: In arithmetic sequences, once you know the first term and the common difference, you can generate any term using the formula for the $n$-th term.