Given the following: a1​=−3, d=2, Sn​=21 , how many terms do the arithmetic sequence have?


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!

Given the following: a1​=−3, d=2, Sn​=21, how many terms do the arithmetic sequence have?

Explore how to determine the number of terms in an arithmetic sequence where the first term is -3, the common difference is 2, and the sum of the sequence is 21. The solution involves using the sum formula for arithmetic sequences and solving a quadratic equation. Suitable for students in Grades 9-10.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation