New question: The sum of the first n terms of an arithmetic sequence is 99. The first term is 3 and the common difference is 3. How many terms are there? - Coaching Toolbox

Understanding the Context

Why This Question Is Resonating Now

Across US schools, math tutors, and online learning platforms, questions like this surface regularly during study sessions and problem-solving workshops. The blend of concrete numbers—3 as the first term and 3 as the common difference—creates a relatable entry point. With more families prioritizing STEM education and digital resources, topics tied to patterns, sums, and data analysis naturally attract engaged learners searching for clarity.

The question also taps into a broader interest in how sequences simplify real-life calculations—whether in budgeting, predicting trends, or understanding growth patterns. As algorithms and data-driven decisions shape industries from finance to tech, knowing how to work with arithmetic sequences enhances critical thinking and pattern recognition skills.

Key Insights

How to Solve: The Exact Answer — Step by Step

Let’s clarify the formula that makes this problem solvable. In an arithmetic sequence, each term follows the rule:
aₙ = a₁ + (n – 1)·d
where a₁ is the first term, d is the common difference, and aₙ is the nth term.

The sum of the first n terms is:
Sₙ = n · (a₁ + aₙ) · ½

We’re told:

• Sₙ = 99
• a₁ = 3
• d = 3

Final Thoughts

First, find aₙ using the sequence rule:
aₙ = 3 + (n – 1)·3 = 3 + 3n – 3 = 3n
Now substitute into the sum formula:
99 = n · (3 + 3n) · ½

Multiply both sides by 2:
198 = n (3 + 3n)
198 = 3n + 3n²
Divide both sides by 3:
66 = n + n²

Rearrange into standard quadratic form:
n² + n – 66 = 0

Use the quadratic formula:
n = [–1 ± √(1 + 264)] / 2 = [–1 ± √265] / 2

Since √265 ≈ 16.28,
n = (–1 + 16.28)/2 ≈ 15.28 / 2 ≈ 7.64

But n must be a whole number because you can’t have a fractional number of terms. Check integer values near 7.64: