The sum of the first 10 terms of an arithmetic sequence is 145, and the first term is 5. Find the common difference. - AIKO, infinite ways to autonomy.

Understanding the Context

Why The Sum of the First 10 Terms Is 145 with First Term 5 Is Gaining Attention Now

In an era where data literacy is increasingly essential, questions like “How do we derive unknown variables in structured sequences?” reflect growing interest in logic-based problem solving. The specific instance—sum of 10 terms equals 145, starting at 5—might seem simple, but its relevance lies in the foundational role arithmetic sequences play in modeling predictable progressions. While not flashy, this problem resonates in practical fields such as algorithmic budgeting, educational performance tracking, and infrastructure planning.

Digital platforms—especially mobile-first search results—respond to user intent rooted in problem-solving and prediction. When users search phrases like “sum of first 10 terms arithmetic sequence 145 first term 5,” they’re often navigating educational needs, practical examples, or even coding challenges. These queries reflect a curious, discerning audience eager for trustworthy, accurate explanations rather than generic summaries. This demand fuels visibility for well-structured, neutral content that illuminates core concepts clearly.

Key Insights

How the Sum of an Arithmetic Sequence Works—and How to Solve It

An arithmetic sequence follows a consistent step difference—called the common difference—between consecutive terms. Given the first term ( a = 5 ), and the sum of the first 10 terms ( S_{10} = 145 ), we use the standard formula for the sum of the first ( n ) terms:

[ S_n = \frac{n}{2} \cdot (2a + (n - 1)d) ]

Plugging in known values:
( n = 10 ), ( S_{10} = 145 ), ( a = 5 ):

Final Thoughts

[ 145 = \frac{10}{2} \cdot (2 \cdot 5 + 9d) ]

Simplify:
[ 145 = 5 \cdot (10 + 9d) ]

[ 145 = 50 + 45d ]

Subtract 50 from both sides:
[ 95 = 45d ]

Solve for ( d ):
[ d = \frac{95}{45} = \frac{19}{9} ]

So the common difference is ( \frac{19}{9} )—a fractional but rational number—showing how sequences can grow or progress by incremental increments rather than whole numbers. This precision matters in modeling real-world processes where gradual change matters, from financial projections to demographic analyses.