Question

Compute \frac{3}{4}\cdot \frac{8}{9}\cdot \frac{15}{16}\cdot \frac{24}{25}\cdot \frac{35}{36}\cdot \frac{48}{49}\cdot \frac{63}{64}\c.... Express your answer in the simplest way possible.

Answer 1

please help

Answer 2

\[\Large\rm \frac{3}{4}\cdot\frac{8}{9}\cdot\frac{15}{16}\cdot\frac{24}{25}\cdot\frac{35}{36}\cdot\frac{48}{49}\cdot\cdot\cdot\]Umm so it looks like we have perfect squares in the bottom starting from 2, yes?

Answer 3

yeah

Answer 4

The top would be the same squares, but with 1 subtracted afterwards.

Answer 5

So the general form of each term looks something like this:\[\Large\rm \frac{n^2-1}{n^2}\]I don't have a lot of experience using the product operator thing...
But I think it would look something like this:\[\Large\rm \prod_{n=2}^{\infty}~\frac{n^2-1}{n^2}\]

Answer 6

ok

Answer 7

Does that symbol look familiar? Have you learned about that?
It tells you to multiply things instead of add.

Answer 8

oh, I haven't learned that

Answer 9

No? Hmm, I'm not sure of another simple way to write it >.<

\[\Large\rm \sum_{n=1}^{3}n\]This symbol tells you to add things: \(\Large\rm 1+2+3\)

\[\Large\rm \prod_{n=1}^3n\]This symbol tells you to multiply things: \(\Large\rm 1\cdot2\cdot3\)



Hopefully it makes a little sense to you :d

Answer 10

Thanks