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Proposition: Sum of Arithmetic Progression

Let \(n\in\mathbb N\) be a natural number and let \(a,b \in F\) be any two elements of a given field \((F, +, \cdot)\). Then the general sum of the arithmetic progression \(a, a + b, a + 2b, a + 3b,\ldots, a + nb \) is

\[S:=\sum_{0\le k\le n} (a+b\cdot k)=(n+1)\cdot \left(a+\frac{bn}2\right).\]

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References

Bibliography

1. Graham L. Ronald, Knuth E. Donald, Patashnik Oren: "Concrete Mathematics", Addison-Wesley, 1994, 2nd Edition