To find the sum of the given arithmetic sequence $-3, 1, 5, 9, 13, \dots, (4n - 7)$, we proceed as follows:

Step 1: Analyze the arithmetic sequence

An arithmetic sequence has a common difference $d$.
Here:

• First term $a = -3$
• Second term $a_2 = 1$

The common difference $d$ is: $d = a_2 - a_1 = 1 - (-3) = 4.$

Step 2: General formula for the $n$-th term

The formula for the $n$-th term of an arithmetic sequence is: $a_n = a + (n-1)d.$ Substitute $a = -3$ and $d = 4$: $a_n = -3 + (n-1)(4).$ Simplify: $a_n = -3 + 4n - 4 = 4n - 7.$ Thus, the $n$-th term is $4n - 7$.

Step 3: Formula for the sum of $n$-terms

The sum $S_n$ of the first $n$-terms in an arithmetic sequence is given by: $S_n = \frac{n}{2} \left( a + a_n \right),$ where $a$ is the first term and $a_n$ is the $n$-th term.

Substitute $a = -3$ and $a_n = 4n - 7$: $S_n = \frac{n}{2} \left( -3 + (4n - 7) \right).$ Simplify inside the parentheses: $S_n = \frac{n}{2} \left( -3 + 4n - 7 \right).$ $S_n = \frac{n}{2} \left( 4n - 10 \right).$ Factor out $2$ from $4n - 10$: $S_n = \frac{n}{2} \cdot 2(2n - 5).$ Simplify: $S_n = n(2n - 5).$

Final Answer:

The sum of the arithmetic sequence is $n(2n - 5)$.

Correct option:
$n(2n - 5)$

Do you need further clarifications or a detailed explanation of any step?

Related Questions:

1. How do you derive the formula for the $n$-th term of an arithmetic sequence?
2. What is the general formula for the sum of $n$-terms of an arithmetic series?
3. How can you identify a common difference in any arithmetic sequence?
4. If the sum of the sequence is $S_n$, how do you solve for $n$ given $S_n$?
5. How do you graph an arithmetic sequence?

Tip:

In arithmetic sequences, always check for the common difference $d$ first, as it helps to derive both the $n$-th term and the sum formula efficiently.