Property 6
a) \(\sum\limits_{k=r}^{n}f(k)=\sum\limits_{k=r-a}^{n-a}f(k+a)\)
b) \(\sum\limits_{k=r}^{n}f(k)=\sum\limits_{k=r+a}^{n+a}f(k-a)\)
c) \(\sum\limits_{k=r}^{n}f(k)=\sum\limits_{k=1}^{n}f(k)-\sum\limits_{k=1}^{r-1}f(k)\)
All the addition rules start from ‘1’ but in some cases where the index does not start from ‘1’, a and b rules are applied to make the index start from ‘1’. Missing terms are found by applying c rule instead of changing the index.
Note: If the function is a simple equation a and b rules are applied. If the function is a quadratic or a cubic equation then c rule is applied.
1.
\(\sum\limits_{k=5}^{24}(2k-1)=?\)
2.
\(\sum\limits_{k=11}^{30}(3k+2)=?\)
3.
\(\sum\limits_{k=-3}^{15}(4k+1)=?\)
4.
\(\sum\limits_{k=-5}^{10}(3k-10)=?\)
5.
\(\sum\limits_{k=7}^{30}(5k+8)=?\)
6.
\(\sum\limits_{k=4}^{17}(2k+5)=?\)
7.
\(\sum\limits_{k=5}^{13}k^2=?\)
8.
\(\sum\limits_{k=6}^{14}k^3=?\)
9.
\(\sum\limits_{k=10}^{20}2^k=?\)
10.
\(\sum\limits_{k=8}^{23}(k\cdot k!)=?\)