Property 9
- \(\prod\limits_{k=1}^{n}c=\underbrace{c\cdot c\cdot c\cdot\cdots\cdot c}_{n\ \text{(n times)}}=c^n\quad(c\in\mathbb R)\)
- \(\prod\limits_{k=r}^{n}c=\underbrace{c\cdot c\cdot c\cdot\cdots\cdot c}_{n-r+1\ \text{(n-r+1 times)}}=c^{n-r+1}\)
1.
\(\prod\limits_{k=1}^{10}2=?\)
2.
\(\prod\limits_{k=2}^{17}3=?\)
3.
\(\prod\limits_{k=0}^{20}7=?\)
4.
\(\prod\limits_{k=10}^{32}2=?\)
5.
\(\prod\limits_{k=6}^{27}5=?\)
6.
\(\prod\limits_{k=9}^{18}1=?\)
7.
\(\prod\limits_{k=15}^{41}(-1)=?\)
Property 10
- \(\prod\limits_{k=r}^{n}[f(k)\cdot g(k)]=\prod\limits_{k=r}^{n}f(k)\prod\limits_{k=r}^{n}g(k)\)
- \(\prod\limits_{k=r}^{n}\samefrac{f(k)}{g(k)}=\samefrac{\prod\limits_{k=1}^{n}f(k)}{\prod\limits_{k=1}^{n}g(k)},\ g(k)\ne0\)
- \(\prod\limits_{k=1}^{n}c\cdot f(k)=c^n\cdot\prod\limits_{k=1}^{n}f(k)\quad(c\in\mathbb R)\)
- \(\prod\limits_{k=r}^{n}c\cdot f(k)=c^{n-r+1}\cdot\prod\limits_{k=r}^{n}f(k)\quad(c\in\mathbb R)\)
1.
\(\prod\limits_{k=1}^{10}(k^2+k)=?\)
2.
\(\prod\limits_{k=1}^{10}(2k)=?\)
3.
\(\prod\limits_{k=3}^{12}(3k)=?\)
4.
\(\prod\limits_{k=1}^{20}\left(\samefrac{k+2}{k+1}\right)=?\)