I'm currently working on geometric sequences but I don't get something really simple about it.
My first question is:
How does (4^n)/[3^(n-1)] become 4[(4/3)^(n-1)]
and how does
(7^n)/[4^(2n-1)] become (4)[(7^n)/(4^2n)].
Thank you!!!
I'm currently working on geometric sequences but I don't get something really simple about it.
My first question is:
How does (4^n)/[3^(n-1)] become 4[(4/3)^(n-1)]
and how does
(7^n)/[4^(2n-1)] become (4)[(7^n)/(4^2n)].
Thank you!!!
Hello, Shelley!
How does .$\displaystyle \frac{4^n}{3^{n-1}}\:\text{ become }\:4\left(\frac{4}{3}\right)^{n-1}$
We have: .$\displaystyle \frac{4^n}{3^{n-1}} \;=\;\frac{4\cdot4^{n-1}}{3^{n-1}} \;=\;4\cdot\frac{4^{n-1}}{3^{n-1}} \;=\;4\left(\frac{4}{3}\right)^{n-1}$
How does .$\displaystyle \frac{7^n}{4^{2n-1}}\:\text{ become }\:4\cdot\frac{7^n}{4^{2n}}$
We have: .$\displaystyle \frac{7^n}{4^{2n-1}} \;=\;\frac{7^n}{4^{2n}\cdot4^{-1}} \;=\;\frac{7^n}{4^{2n}\cdot\frac{1}{4}} \;=\;4\cdot\frac{7^n}{4^{2n}}$
Ha! . . . We can push it even further . . .
We have: .$\displaystyle 4\cdot\frac{7^n}{(4^2)^n} \;=\;4\cdot\frac{7^n}{16^n} \;=\;7\left(\frac{7}{16}\right)^n
$