Answer:
Show us the equation
Step-by-step explanation:
Could you put a screenshot of it
0.092, 0.4, 0.432, 0.821 least to greatest so 0.092 is least
For starters,
![\dfrac{3^k}{4^{k+2}}=\dfrac{3^k}{4^24^k}=\dfrac1{16}\left(\dfrac34\right)^k](https://tex.z-dn.net/?f=%5Cdfrac%7B3%5Ek%7D%7B4%5E%7Bk%2B2%7D%7D%3D%5Cdfrac%7B3%5Ek%7D%7B4%5E24%5Ek%7D%3D%5Cdfrac1%7B16%7D%5Cleft%28%5Cdfrac34%5Cright%29%5Ek)
Consider the
th partial sum, denoted by
:
![S_n=\dfrac1{16}\left(\dfrac34\right)+\dfrac1{16}\left(\dfrac34\right)^2+\dfrac1{16}\left(\dfrac34\right)^3+\cdots+\dfrac1{16}\left(\dfrac34\right)^n](https://tex.z-dn.net/?f=S_n%3D%5Cdfrac1%7B16%7D%5Cleft%28%5Cdfrac34%5Cright%29%2B%5Cdfrac1%7B16%7D%5Cleft%28%5Cdfrac34%5Cright%29%5E2%2B%5Cdfrac1%7B16%7D%5Cleft%28%5Cdfrac34%5Cright%29%5E3%2B%5Ccdots%2B%5Cdfrac1%7B16%7D%5Cleft%28%5Cdfrac34%5Cright%29%5En)
Multiply both sides by
:
![\dfrac34S_n=\dfrac1{16}\left(\dfrac34\right)^2+\dfrac1{16}\left(\dfrac34\right)^3+\dfrac1{16}\left(\dfrac34\right)^4+\cdots+\dfrac1{16}\left(\dfrac34\right)^{n+1}](https://tex.z-dn.net/?f=%5Cdfrac34S_n%3D%5Cdfrac1%7B16%7D%5Cleft%28%5Cdfrac34%5Cright%29%5E2%2B%5Cdfrac1%7B16%7D%5Cleft%28%5Cdfrac34%5Cright%29%5E3%2B%5Cdfrac1%7B16%7D%5Cleft%28%5Cdfrac34%5Cright%29%5E4%2B%5Ccdots%2B%5Cdfrac1%7B16%7D%5Cleft%28%5Cdfrac34%5Cright%29%5E%7Bn%2B1%7D)
Subtract
from this:
![\dfrac34S_n-S_n=\dfrac1{16}\left(\dfrac34\right)^{n+1}-\dfrac1{16}\left(\dfrac34\right)](https://tex.z-dn.net/?f=%5Cdfrac34S_n-S_n%3D%5Cdfrac1%7B16%7D%5Cleft%28%5Cdfrac34%5Cright%29%5E%7Bn%2B1%7D-%5Cdfrac1%7B16%7D%5Cleft%28%5Cdfrac34%5Cright%29)
Solve for
:
![-\dfrac14S_n=\dfrac3{64}\left(\left(\dfrac34\right)^n-1\right)](https://tex.z-dn.net/?f=-%5Cdfrac14S_n%3D%5Cdfrac3%7B64%7D%5Cleft%28%5Cleft%28%5Cdfrac34%5Cright%29%5En-1%5Cright%29)
![S_n=\dfrac3{16}\left(1-\left(\dfrac34\right)^n\right)](https://tex.z-dn.net/?f=S_n%3D%5Cdfrac3%7B16%7D%5Cleft%281-%5Cleft%28%5Cdfrac34%5Cright%29%5En%5Cright%29)
Now as
, the exponential term will converge to 0, since
if
. This leaves us with
![\displaystyle\lim_{n\to\infty}S_n=\lim_{n\to\infty}\sum_{k=1}^n\frac{3^k}{4^{k+2}}=\sum_{k=1}^\infty\frac{3^k}{4^{k+2}}=\frac3{16}](https://tex.z-dn.net/?f=%5Cdisplaystyle%5Clim_%7Bn%5Cto%5Cinfty%7DS_n%3D%5Clim_%7Bn%5Cto%5Cinfty%7D%5Csum_%7Bk%3D1%7D%5En%5Cfrac%7B3%5Ek%7D%7B4%5E%7Bk%2B2%7D%7D%3D%5Csum_%7Bk%3D1%7D%5E%5Cinfty%5Cfrac%7B3%5Ek%7D%7B4%5E%7Bk%2B2%7D%7D%3D%5Cfrac3%7B16%7D)
The profit for 25 products sold and 150 products sold are 3159 and 19190.25 respectively.
<u><em>Explanation</em></u>
The profit of a company receives is given by the expression: ![0.15(855p - 315)](https://tex.z-dn.net/?f=0.15%28855p%20-%20315%29)
Simplifying this expression using distributive property, we will get .....
![0.15(855p - 315) \\ \\ = (0.15*855p)-(0.15*315)\\ \\ =128.25p-47.25](https://tex.z-dn.net/?f=0.15%28855p%20-%20315%29%20%5C%5C%20%5C%5C%20%3D%20%280.15%2A855p%29-%280.15%2A315%29%5C%5C%20%5C%5C%20%3D128.25p-47.25)
So, the simplified expression for profit will be: ![128.25p-47.25](https://tex.z-dn.net/?f=128.25p-47.25)
As
represents the number of products sold, so for finding the profit for 25 products sold and 150 products sold, <u>we need plug
and
separately into the above expression</u>.
For
, Profit ![=128.25(25)-47.25 = 3206.25-47.25= 3159](https://tex.z-dn.net/?f=%3D128.25%2825%29-47.25%20%3D%203206.25-47.25%3D%203159)
For
, Profit ![=128.25(150)-47.25 = 19237.50-47.25= 19190.25](https://tex.z-dn.net/?f=%3D128.25%28150%29-47.25%20%3D%2019237.50-47.25%3D%2019190.25)