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Solution

(related to Problem: Sum of Consecutive Positive Integers)

$$1+2+3+\ldots+n=\sum_{x=1}^n x^{\underline{1}}=\frac{x(x-1)}{2}\;\Rule{1px}{4ex}{2ex}^{n+1}_{1}=\frac{(n+1)n}{2}.$$

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References

Bibliography

1. Miller, Kenneth S.: "An Introduction to the Calculus of Finite Differences And Difference Equations", Dover Publications, Inc, 1960