The sum of the first n terms of an arithmetic sequence is 210. If the first term is 5 and the common difference is 3, find n. - Sterling Industries
The Sum of the First n Terms of an Arithmetic Sequence Is 210. If the First Term Is 5 and the Common Difference Is 3, Find n.
The Sum of the First n Terms of an Arithmetic Sequence Is 210. If the First Term Is 5 and the Common Difference Is 3, Find n.
Curious about how math brings logic to everyday patterns? The idea that a sequence of numbers follows precise rules captivates learners, educators, and problem-solvers alike—especially in an age where data and structure shape daily decisions. One classic question still sparks interest: What value of n makes the sum of the first n terms of an arithmetic sequence equal 210, starting at 5 with a common difference of 3? This isn’t just an academic puzzle—it reflects how logic applies beyond classrooms, fueling clarity in personal finance, project planning, and trend analysis.
Why The Sum of the First n Terms Is Sparking Curiosity Across the U.S.
Understanding the Context
This problem taps into a broader cultural appetite for understanding numerical relationships—a skill increasingly valuable in a data-driven economy. Whether for school projects, coding challenges, or practical budget modeling, recognizing arithmetic sequences helps break complex problems into manageable parts. Educational platforms and DIY math communities report growing interest in foundational sequences, driven by a desire for analytical confidence and real-world applicability. As digital tools reshape learning, topics like this reinforce structured thinking, making them timeless yet freshly relevant to U.S. audiences navigating constant change.
How the Sum of the First n Terms Actually Works
The formula to calculate the sum of the first n terms in an arithmetic sequence is elegant in its simplicity. Given a sequence where the first term is a₁ = 5 and each step increases by a common difference d = 3, the nth term and total sum depend on arithmetic progression rules.
The general formula for the sum is:
Sₙ = n/2 × [2a₁ + (n – 1)d]
Plugging in the known values:
Sₙ = 210
a₁ = 5
d = 3
Key Insights
Substitute:
210 = n/2 × [2(5) + (n – 1)(3)]
210 = n/2 × [10 + 3n – 3]
210 = n/2 × (7 + 3n)
Multiply both sides by 2:
420 = n(7 + 3n)
420 = 7n + 3n²
Rearranged:
3n² + 7n – 420 = 0
Now solve this quadratic equation using
