Finite sums formulas are shortcuts for adding common patterns of numbers without writing out every term. The most common formulas include sums of constants, integers, squares, and cubes. These problems practice choosing the correct formula, substituting the upper limit, and simplifying the result.
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
\(\textbf{1)}\)\( \displaystyle \sum_{n=1}^{8} n^3 \)
\(1296\)
\(\,\,\,\,\,\displaystyle \sum_{k=1}^{n} k^3 = \displaystyle\frac{n^2(n+1)^2}{4}\)
\(\,\,\,\,\,\displaystyle \sum_{n=1}^{8} n^3 = \displaystyle\frac{(8)^2((8)+1)^2}{4}\)
\(\,\,\,\,\,\displaystyle \sum_{n=1}^{8} n^3 = \displaystyle\frac{64(9)^2}{4}\)
\(\,\,\,\,\,\displaystyle \sum_{n=1}^{8} n^3 = \displaystyle\frac{5184}{4}\)
\(\,\,\,\,\,\displaystyle \sum_{n=1}^{8} n^3 = 1296\)
\(\textbf{2)}\)\( \displaystyle \sum_{n=1}^{6} n^2 \)
\(91\)
\(\,\,\,\,\,\displaystyle \sum_{k=1}^{n} k^2 = \displaystyle\frac{n(n+1)(2n+1)}{6}\)
\(\,\,\,\,\,\displaystyle \sum_{n=1}^{6} n^2 = \displaystyle\frac{(6)((6)+1)(2(6)+1)}{6}\)
\(\,\,\,\,\,\displaystyle \sum_{n=1}^{6} n^2 = \displaystyle\frac{(6)(7)(13)}{6}\)
\(\,\,\,\,\,\displaystyle \sum_{n=1}^{6} n^2 = 91\)
\(\textbf{3)}\)\( \displaystyle \sum_{n=1}^{4} 7 \)
\(28\)
\(\,\,\,\,\,\displaystyle \sum_{k=1}^{n} c = cn\)
\(\,\,\,\,\,\displaystyle \sum_{n=1}^{4} 7 = (4)(7)\)
\(\,\,\,\,\,\displaystyle \sum_{n=1}^{4} 7 = 28\)
\(\textbf{4)}\)\( \displaystyle \sum_{n=1}^{10} n \)
\(55\)
\(\,\,\,\,\,\displaystyle \sum_{k=1}^{n} k = \displaystyle\frac{n(n+1)}{2}\)
\(\,\,\,\,\,\displaystyle \sum_{n=1}^{10} n = \displaystyle\frac{10(10+1)}{2}\)
\(\,\,\,\,\,\displaystyle \sum_{n=1}^{10} n = \displaystyle\frac{10(11)}{2}\)
\(\,\,\,\,\,\displaystyle \sum_{n=1}^{10} n = 55\)
\(\textbf{5)}\)\( \displaystyle \sum_{n=1}^{12} n \)
\(78\)
\(\,\,\,\,\,\displaystyle \sum_{k=1}^{n} k = \displaystyle\frac{n(n+1)}{2}\)
\(\,\,\,\,\,\displaystyle \sum_{n=1}^{12} n = \displaystyle\frac{12(12+1)}{2}\)
\(\,\,\,\,\,\displaystyle \sum_{n=1}^{12} n = \displaystyle\frac{12(13)}{2}\)
\(\,\,\,\,\,\displaystyle \sum_{n=1}^{12} n = 78\)
\(\textbf{6)}\)\( \displaystyle \sum_{n=1}^{9} n^2 \)
\(285\)
\(\,\,\,\,\,\displaystyle \sum_{k=1}^{n} k^2 = \displaystyle\frac{n(n+1)(2n+1)}{6}\)
\(\,\,\,\,\,\displaystyle \sum_{n=1}^{9} n^2 = \displaystyle\frac{9(9+1)(2(9)+1)}{6}\)
\(\,\,\,\,\,\displaystyle \sum_{n=1}^{9} n^2 = \displaystyle\frac{9(10)(19)}{6}\)
\(\,\,\,\,\,\displaystyle \sum_{n=1}^{9} n^2 = 285\)
\(\textbf{7)}\)\( \displaystyle \sum_{n=1}^{5} n^3 \)
\(225\)
\(\,\,\,\,\,\displaystyle \sum_{k=1}^{n} k^3 = \displaystyle\frac{n^2(n+1)^2}{4}\)
\(\,\,\,\,\,\displaystyle \sum_{n=1}^{5} n^3 = \displaystyle\frac{5^2(5+1)^2}{4}\)
\(\,\,\,\,\,\displaystyle \sum_{n=1}^{5} n^3 = \displaystyle\frac{25(36)}{4}\)
\(\,\,\,\,\,\displaystyle \sum_{n=1}^{5} n^3 = 225\)
\(\textbf{8)}\)\( \displaystyle \sum_{n=1}^{15} 4 \)
\(60\)
\(\,\,\,\,\,\displaystyle \sum_{k=1}^{n} c = cn\)
\(\,\,\,\,\,\displaystyle \sum_{n=1}^{15} 4 = 4(15)\)
\(\,\,\,\,\,\displaystyle \sum_{n=1}^{15} 4 = 60\)
\(\textbf{9)}\)\( \displaystyle \sum_{n=1}^{7} n^2 \)
\(140\)
\(\,\,\,\,\,\displaystyle \sum_{k=1}^{n} k^2 = \displaystyle\frac{n(n+1)(2n+1)}{6}\)
\(\,\,\,\,\,\displaystyle \sum_{n=1}^{7} n^2 = \displaystyle\frac{7(7+1)(2(7)+1)}{6}\)
\(\,\,\,\,\,\displaystyle \sum_{n=1}^{7} n^2 = \displaystyle\frac{7(8)(15)}{6}\)
\(\,\,\,\,\,\displaystyle \sum_{n=1}^{7} n^2 = 140\)
\(\textbf{10)}\)\( \displaystyle \sum_{n=1}^{20} n \)
\(210\)
\(\,\,\,\,\,\displaystyle \sum_{k=1}^{n} k = \displaystyle\frac{n(n+1)}{2}\)
\(\,\,\,\,\,\displaystyle \sum_{n=1}^{20} n = \displaystyle\frac{20(20+1)}{2}\)
\(\,\,\,\,\,\displaystyle \sum_{n=1}^{20} n = \displaystyle\frac{20(21)}{2}\)
\(\,\,\,\,\,\displaystyle \sum_{n=1}^{20} n = 210\)
\(\textbf{11)}\)\( \displaystyle \sum_{n=1}^{4} n^3 \)
\(100\)
\(\,\,\,\,\,\displaystyle \sum_{k=1}^{n} k^3 = \displaystyle\frac{n^2(n+1)^2}{4}\)
\(\,\,\,\,\,\displaystyle \sum_{n=1}^{4} n^3 = \displaystyle\frac{4^2(4+1)^2}{4}\)
\(\,\,\,\,\,\displaystyle \sum_{n=1}^{4} n^3 = \displaystyle\frac{16(25)}{4}\)
\(\,\,\,\,\,\displaystyle \sum_{n=1}^{4} n^3 = 100\)
\(\textbf{12)}\)\( \displaystyle \sum_{n=1}^{11} 6 \)
\(66\)
\(\,\,\,\,\,\displaystyle \sum_{k=1}^{n} c = cn\)
\(\,\,\,\,\,\displaystyle \sum_{n=1}^{11} 6 = 6(11)\)
\(\,\,\,\,\,\displaystyle \sum_{n=1}^{11} 6 = 66\)
\(\textbf{13)}\)\( \displaystyle \sum_{n=1}^{10} n^2 \)
\(385\)
\(\,\,\,\,\,\displaystyle \sum_{k=1}^{n} k^2 = \displaystyle\frac{n(n+1)(2n+1)}{6}\)
\(\,\,\,\,\,\displaystyle \sum_{n=1}^{10} n^2 = \displaystyle\frac{10(10+1)(2(10)+1)}{6}\)
\(\,\,\,\,\,\displaystyle \sum_{n=1}^{10} n^2 = \displaystyle\frac{10(11)(21)}{6}\)
\(\,\,\,\,\,\displaystyle \sum_{n=1}^{10} n^2 = 385\)
\(\textbf{14)}\)\( \displaystyle \sum_{n=1}^{3} n^3 \)
\(36\)
\(\,\,\,\,\,\displaystyle \sum_{k=1}^{n} k^3 = \displaystyle\frac{n^2(n+1)^2}{4}\)
\(\,\,\,\,\,\displaystyle \sum_{n=1}^{3} n^3 = \displaystyle\frac{3^2(3+1)^2}{4}\)
\(\,\,\,\,\,\displaystyle \sum_{n=1}^{3} n^3 = \displaystyle\frac{9(16)}{4}\)
\(\,\,\,\,\,\displaystyle \sum_{n=1}^{3} n^3 = 36\)
\(\textbf{15)}\)\( \displaystyle \sum_{n=1}^{25} n \)
\(325\)
\(\,\,\,\,\,\displaystyle \sum_{k=1}^{n} k = \displaystyle\frac{n(n+1)}{2}\)
\(\,\,\,\,\,\displaystyle \sum_{n=1}^{25} n = \displaystyle\frac{25(25+1)}{2}\)
\(\,\,\,\,\,\displaystyle \sum_{n=1}^{25} n = \displaystyle\frac{25(26)}{2}\)
\(\,\,\,\,\,\displaystyle \sum_{n=1}^{25} n = 325\)
Challenge Problems
\(\textbf{16)}\)\( \displaystyle \sum_{n=1}^{8} (3n+2) \)
\(124\)
\(\,\,\,\,\,\displaystyle \sum_{n=1}^{8} (3n+2)=3\displaystyle\sum_{n=1}^{8}n+\displaystyle\sum_{n=1}^{8}2\)
\(\,\,\,\,\,3\displaystyle\sum_{n=1}^{8}n=3\left(\displaystyle\frac{8(8+1)}{2}\right)=3(36)=108\)
\(\,\,\,\,\,\displaystyle\sum_{n=1}^{8}2=2(8)=16\)
\(\,\,\,\,\,108+16=124\)
\(\textbf{17)}\)\( \displaystyle \sum_{n=1}^{5} (n^2+4) \)
\(75\)
\(\,\,\,\,\,\displaystyle \sum_{n=1}^{5}(n^2+4)=\displaystyle\sum_{n=1}^{5}n^2+\displaystyle\sum_{n=1}^{5}4\)
\(\,\,\,\,\,\displaystyle\sum_{n=1}^{5}n^2=\displaystyle\frac{5(5+1)(2(5)+1)}{6}=\frac{5(6)(11)}{6}=55\)
\(\,\,\,\,\,\displaystyle\sum_{n=1}^{5}4=4(5)=20\)
\(\,\,\,\,\,55+20=75\)
\(\textbf{18)}\)\( \displaystyle \sum_{n=1}^{6} (2n^2-n) \)
\(161\)
\(\,\,\,\,\,\displaystyle \sum_{n=1}^{6}(2n^2-n)=2\displaystyle\sum_{n=1}^{6}n^2-\displaystyle\sum_{n=1}^{6}n\)
\(\,\,\,\,\,2\displaystyle\sum_{n=1}^{6}n^2=2(91)=182\)
\(\,\,\,\,\,\displaystyle\sum_{n=1}^{6}n=\displaystyle\frac{6(6+1)}{2}=21\)
\(\,\,\,\,\,182-21=161\)
\(\textbf{19)}\)\( \displaystyle \sum_{n=1}^{4} (n^3+n^2) \)
\(130\)
\(\,\,\,\,\,\displaystyle \sum_{n=1}^{4}(n^3+n^2)=\displaystyle\sum_{n=1}^{4}n^3+\displaystyle\sum_{n=1}^{4}n^2\)
\(\,\,\,\,\,\displaystyle\sum_{n=1}^{4}n^3=100\)
\(\,\,\,\,\,\displaystyle\sum_{n=1}^{4}n^2=\displaystyle\frac{4(4+1)(2(4)+1)}{6}=30\)
\(\,\,\,\,\,100+30=130\)
\(\textbf{20)}\)\( \displaystyle \sum_{n=1}^{10} (5n-3) \)
\(245\)
\(\,\,\,\,\,\displaystyle \sum_{n=1}^{10}(5n-3)=5\displaystyle\sum_{n=1}^{10}n-\displaystyle\sum_{n=1}^{10}3\)
\(\,\,\,\,\,5\displaystyle\sum_{n=1}^{10}n=5\left(\displaystyle\frac{10(10+1)}{2}\right)=5(55)=275\)
\(\,\,\,\,\,\displaystyle\sum_{n=1}^{10}3=3(10)=30\)
\(\,\,\,\,\,275-30=245\)
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