Property 6
\(\sum_{k=r}^{n}f(k)=\sum_{k=r-a}^{n-a}f(k+a)\)
\(\sum_{k=r}^{n}f(k)=\sum_{k=r+a}^{n+a}f(k-a)\)
\(\sum_{k=r}^{n}f(k)=\sum_{k=1}^{n}f(k)-\sum_{k=1}^{r-1}f(k)\)
When the index does not start at 1, shift the index or subtract the missing initial terms. For linear functions the first two rules are convenient; for quadratic or cubic functions the subtraction rule is often used.
1.
\(\sum_{k=5}^{24}(2k-1)=?\)
2.
\(\sum_{k=11}^{30}(3k+2)=?\)
3.
\(\sum_{k=-3}^{15}(4k+1)=?\)
4.
\(\sum_{k=-5}^{10}(3k-10)=?\)
5.
\(\sum_{k=7}^{30}(5k+8)=?\)
6.
\(\sum_{k=4}^{17}(2k+5)=?\)
7.
\(\sum_{k=5}^{13}k^2=?\)
8.
\(\sum_{k=6}^{14}k^3=?\)
9.
\(\sum_{k=10}^{20}2^k=?\)
10.
\(\sum_{k=8}^{23}(k\cdot k!)=?\)