The problem asks for a formula for the sum of the first three terms in an arithmetic sequence, using only the variables $a_1$ (the first term) and $d$ (the common difference).

In an arithmetic sequence, the terms can be written as:

• First term: $a_1$
• Second term: $a_2 = a_1 + d$
• Third term: $a_3 = a_1 + 2d$

Now, the sum of the first three terms is: $S = a_1 + a_2 + a_3$ Substitute the expressions for $a_2$ and $a_3$: $S = a_1 + (a_1 + d) + (a_1 + 2d)$ Simplify the expression: $S = a_1 + a_1 + d + a_1 + 2d = 3a_1 + 3d$ Thus, the formula for the sum of the first three terms is: $S = 3a_1 + 3d$

Would you like further clarification on this?

Here are some related questions for practice:

1. What is the formula for the sum of the first four terms in an arithmetic sequence?
2. How would the formula change if we wanted the sum of the first five terms?
3. What is the general formula for the sum of the first $n$ terms in an arithmetic sequence?
4. If $a_1 = 5$ and $d = 2$, what is the sum of the first three terms?
5. How does the sum change if the common difference $d$ is negative?

Tip: Always verify the terms of an arithmetic sequence by checking the common difference $d$ between consecutive terms.