To find the sum of the first 10 terms of the sequence where the first term $a_1 = 2$ and each subsequent term increases by $d = 3$, we use the formula for the sum of an arithmetic sequence:

$S_n = \frac{n}{2} \cdot (a_1 + a_n)$

Step 1: Calculate the 10th term ($a_{10}$)

The formula for the $n$-th term of an arithmetic sequence is:

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

Substitute $a_1 = 2$, $d = 3$, and $n = 10$:

$a_{10} = 2 + (10-1) \cdot 3 = 2 + 27 = 29$

Step 2: Calculate the sum of the first 10 terms ($S_{10}$)

Substitute $n = 10$, $a_1 = 2$, and $a_{10} = 29$ into the sum formula:

$S_{10} = \frac{10}{2} \cdot (2 + 29) = 5 \cdot 31 = 155$

Final Answer:

The sum of the first 10 terms is:

$\boxed{155}$

Would you like further explanation or related examples? Here are five related questions for practice:

1. Find the sum of the first 20 terms of the same sequence.
2. What is the sum of the first 10 terms of the sequence if the common difference is 5 instead of 3?
3. Derive the general formula for the sum of the first $n$ terms of this sequence.
4. Calculate the 15th term of this sequence.
5. What is the difference between the sum of the first 10 terms and the first 5 terms?

Tip: Always double-check arithmetic sequences by writing out the first few terms to ensure the common