I have 2 sequences I have to represent with a Product symbol (Separately of course).
a) $2*\sqrt{3}*\sqrt{4}^3*\sqrt{5}^4$
b) $\dfrac{1}{n}*\dfrac{3}{n+2}*\dfrac{7}{n+3}$
I am clueless on what I should use as a start and end variable. $$\prod_{i=0}^n i$$
For A I was thinking that it has something to do with 2, since you need 2 in the exponent to remove the square root.
Any suggestions?
Products are like sums (and similar to integrals) in that the name of the index variable does not matter.
In your first sequence, the terms are $2^1=2^{2/2},3^{1/2},4^{3/2},5^{4/2}$, I cannot obviously detect a pattern. But simpler one can serve as an example, say $2^1,3^{1/2},4^{1/3},5^{1/4}$, etc. It's easy to see the denominator in the exponent is one less than the base, and the base goes from 2 to 5. So you get $$ \prod_{k=2}^5 k^{1/(k-1)} = \prod_{k=1}^4 (k+1)^{1/k} $$
Can you now try both sequences after checking the first one for correctness?