To find the number of terms $n$ in an arithmetic sequence given:

• First term $a_1 = -3$
• Common difference $d = 2$
• Sum of the first $n$ terms $S_n = 21$

we use the formula for the sum of the first $n$ terms of an arithmetic sequence:

$S_n = \frac{n}{2} \left(2a_1 + (n - 1)d \right)$

Substitute the given values into the formula:

$21 = \frac{n}{2} \left(2(-3) + (n - 1) \cdot 2\right)$

Simplify the expression inside the parentheses:

$21 = \frac{n}{2} \left(-6 + 2(n - 1)\right)$

$21 = \frac{n}{2} \left(-6 + 2n - 2\right)$

$21 = \frac{n}{2} \left(2n - 8\right)$

Multiply both sides by 2 to eliminate the fraction:

$42 = n(2n - 8)$

Expand and simplify:

$42 = 2n^2 - 8n$

Rearrange the equation to form a standard quadratic equation:

$2n^2 - 8n - 42 = 0$

Divide the entire equation by 2:

$n^2 - 4n - 21 = 0$

Now, solve for $n$ using the quadratic formula:

$n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

where $a = 1$, $b = -4$, and $c = -21$. Substitute these values into the formula:

$n = \frac{-(-4) \pm \sqrt{(-4)^2 - 4 \cdot 1 \cdot (-21)}}{2 \cdot 1}$

Simplify:

$n = \frac{4 \pm \sqrt{16 + 84}}{2}$

$n = \frac{4 \pm \sqrt{100}}{2}$

$n = \frac{4 \pm 10}{2}$

This gives two possible solutions:

$n = \frac{4 + 10}{2} = \frac{14}{2} = 7$

$n = \frac{4 - 10}{2} = \frac{-6}{2} = -3$

Since the number of terms $n$ must be a positive integer, we discard $n = -3$.

Thus, the arithmetic sequence has 7 terms.

Would you like more details or have any questions?

Here are 5 related questions to explore further:

1. How do you find the nth term of an arithmetic sequence?
2. What is the sum formula for a geometric sequence?
3. How do you derive the quadratic formula from a general quadratic equation?
4. What are some real-world applications of arithmetic sequences?
5. How do you solve quadratic equations by completing the square?

Tip: When working with sequences, always check if the number of terms can realistically match the context of the problem!