Evaluate some finite sums

by RoRi

July 4, 2015

$\displaystyle{\sum_{k=1}^4 k = 1 + 2 + 3 + 4 = 10}.$
$\displaystyle{\sum_{n=2}^5 2^{n-2} = 2^0 + 2^1 + 2^2 + 2^3 = 1 + 2 + 4 + 8 = 15}.$
$\displaystyle{\sum_{r=0}^3 2^{2r+1} = 2^1 + 2^3 + 2^5 + 2^7 = 2 + 8 + 32 + 128 = 170}.$
$\displaystyle{\sum_{n=1}^4 n^n = 1^1 + 2^2 + 3^3 + 4^4 = 1 + 4 + 27 + 256 = 288}.$
$\displaystyle{\sum_{i=0}^5 (2i+1) = 1 + 3 + 5 + 7 + 9 + 11 = 36}.$
$\displaystyle{\sum_{k=1}^5 \frac{1}{k(k+1)} = \frac{1}{2} + \frac{1}{6} + \frac{1}{12} + \frac{1}{20} + \frac{1}{30} = \frac{5}{6}}.$

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