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Gauss's pairing trick sums 1 to n by pairing the first and last terms: each pair totals n + 1 and there are n/2 pairs, giving n(n + 1)/2.
Triangular sum
1 + 2 + … + n = n(n + 1) / 2
100 × 101 / 2 = 5050, the sum the young Gauss reportedly found in seconds.
Because the terms are evenly spaced, so the mean is the midpoint of the first and last term.