We are tasked with finding the sum of the first 36 terms of the arithmetic sequence $7, 12, 17, \ldots$. Let us solve step by step.

Step 1: Identify the terms of the arithmetic sequence

• The first term ($a$) is $7$.
• The common difference ($d$) is $12 - 7 = 5$.

Step 2: Use the formula for the sum of the first $n$ terms of an arithmetic sequence:

$S_n = \frac{n}{2} \cdot \left(2a + (n - 1)d\right)$ Here:

• $n = 36$,
• $a = 7$,
• $d = 5$.

Substitute the values: $S_{36} = \frac{36}{2} \cdot \left(2(7) + (36 - 1)(5)\right)$

Step 3: Simplify

1. Calculate $2a = 2 \times 7 = 14$,
2. Calculate $(n - 1)d = (36 - 1)(5) = 35 \times 5 = 175$,
3. Add $2a + (n - 1)d = 14 + 175 = 189$,
4. Multiply by $\frac{n}{2}$: $S_{36} = 18 \times 189 = 3402.$

Final Answer:

The sum of the first 36 terms is approximately 3402, rounded to the nearest integer.

Would you like a detailed breakdown of any step? Let me know! Here are some follow-up questions:

1. How is the formula for the sum of an arithmetic sequence derived?
2. What happens if the sequence has a fractional or negative common difference?
3. Can we use the same formula for geometric sequences?
4. How do we find the 36th term of this arithmetic sequence?
5. How does the sum formula change if the series starts with a different number?

Tip: Always check the type of sequence (arithmetic or geometric) before applying a formula!