Notes
\( \displaystyle \sum_{k=1}^{n} c = cn \) (Sum of Constants Formula)
\( \displaystyle \sum_{k=1}^{n} k = \displaystyle\frac{n(n+1)}{2} \) (Sum of Integers Formula)
\( \displaystyle \sum_{k=1}^{n} k^2 = \displaystyle\frac{n(n+1)(2n+1)}{6} \) (Sum of Squares Formula)
\( \displaystyle \sum_{k=1}^{n} k^3 = \displaystyle\frac{n^2(n+1)^2}{4} \) (Sum of Cubes Formula)
Practice Problems
\(\textbf{1)}\)\( \displaystyle \sum_{n=1}^{8} n^3 \)
\(1296\)
\(\,\,\,\,\,\displaystyle \sum_{k=1}^{n} k^3 = \displaystyle\frac{n^2(n+1)^2}{4}\)
\(\,\,\,\,\,\displaystyle \sum_{n=1}^{8} n^3 = \displaystyle\frac{(8)^2((8)+1)^2}{4}\)
\(\,\,\,\,\,\displaystyle \sum_{n=1}^{8} n^3 = \displaystyle\frac{64(9)^2}{4}\)
\(\,\,\,\,\,\displaystyle \sum_{n=1}^{8} n^3 = \displaystyle\frac{5184}{4}\)
\(\,\,\,\,\,\displaystyle \sum_{n=1}^{8} n^3 = 1296\)
\(\textbf{2)}\)\( \displaystyle \sum_{n=1}^{6} n^2 \)
\(91\)
\(\,\,\,\,\,\displaystyle \sum_{k=1}^{n} k^2 = \displaystyle\frac{n(n+1)(2n+1)}{6}\)
\(\,\,\,\,\,\displaystyle \sum_{n=1}^{6} n^2 = \displaystyle\frac{(6)((6)+1)(2(6)+1)}{6}\)
\(\,\,\,\,\,\displaystyle \sum_{n=1}^{6} n^2 = \displaystyle\frac{(6)(7)(13)}{6}\)
\(\,\,\,\,\,\displaystyle \sum_{n=1}^{6} n^2 = 91\)
\(\textbf{3)}\)\( \displaystyle \sum_{n=1}^{4} 7 \)
\(28\)
\(\,\,\,\,\,\displaystyle \sum_{k=1}^{n} c = cn\)
\(\,\,\,\,\,\displaystyle \sum_{n=1}^{4} 7 = (4)(7)\)
\(\,\,\,\,\,\displaystyle \sum_{n=1}^{4} 7 = 28\)
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