Summing Linear Sequences: From Gauss to Modern Mathematics
Calculate the sum of arithmetic sequences using the elegant formula Sₙ = n/2 × (a₁ + aₙ). Learn Gauss's method and applications.
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Summing a linear (arithmetic) sequence is one of the oldest problems in mathematics. The elegant formula Sₙ = n/2 × (a₁ + aₙ) — famously attributed to young Carl Friedrich Gauss — provides a direct way to compute the sum without adding each term individually.
The Formula
Where n is the number of terms, a₁ is the first term, and aₙ is the last term.
Gauss\'s Method
The young Gauss discovered that you can pair terms from opposite ends of the sequence. Each pair sums to the same value (a₁ + aₙ), and there are n/2 pairs.
Worked Example
Sum of 1 + 2 + 3 + ... + 100:
- a₁ = 1, aₙ = 100, n = 100
- S₁₀₀ = 100/2 × (1 + 100) = 50 × 101 = 5050
Applications
- Financial planning: Total savings with regular deposits
- Physics: Distance traveled under uniform acceleration
- Computer science: Analysis of loop iterations