Property 11
- \(\prod\limits_{k=r}^{n}f(k)=\samefrac{\prod\limits_{k=1}^{n}f(k)}{\prod\limits_{k=1}^{r-1}f(k)}\)
- \(\prod\limits_{k=r}^{n}f(k)=\prod\limits_{k=r+a}^{n+a}f(k-a)\)
- \(\prod\limits_{k=r}^{n}f(k)=\prod\limits_{k=r-a}^{n-a}f(k+a)\)
1.
\(\prod\limits_{k=3}^{10}(k-2)=?\)
2.
\(\prod\limits_{k=-4}^{17}(k+5)=?\)
3.
\(\prod\limits_{k=-3}^{10}(2k+8)=?\)
4.
\(\prod\limits_{k=-5}^{12}(3k+18)=?\)
5.
\(\prod\limits_{k=-2}^{11}(k^2-1)=?\)
Property 12
- \(\begin{aligned}\prod\limits_{k=r}^{n}c^{f(k)}&=c^{f(r)}\cdot c^{f(r+1)}\cdot\cdots\cdot c^{f(n)}\\&=c^{f(r)+f(r+1)+\cdots+f(n)}\\&=c^{\left(\sum\limits_{k=r}^{n}f(k)\right)}\end{aligned}\)
\(\prod\limits_{k=r}^{n}c^{f(k)}=c^{\left(\sum\limits_{k=r}^{n}f(k)\right)}\)
1.
\(\prod\limits_{k=1}^{10}2^k=?\)
2.
\(\prod\limits_{k=4}^{20}\left(3^{k-3}\right)=?\)
3.
\(\prod\limits_{k=1}^{15}\left(\samefrac{1}{4^k}\right)=?\)
4.
\(\prod\limits_{k=2}^{12}\left(3^{k^2-1}\right)=?\)
5.
\(\prod\limits_{k=1}^{11}\sqrt{6^k}=?\)