Answer:
The next fraction in the given geometric sequence is ![\frac{40}{243}](https://tex.z-dn.net/?f=%5Cfrac%7B40%7D%7B243%7D)
Therefore ![a_5=\frac{40}{243}](https://tex.z-dn.net/?f=a_5%3D%5Cfrac%7B40%7D%7B243%7D)
Step-by-step explanation:
Given geometric sequence is
![\frac{5}{6}, \frac{5}{9}, \frac{10}{27}, \frac{20}{81}, \ldots](https://tex.z-dn.net/?f=%5Cfrac%7B5%7D%7B6%7D%2C%20%5Cfrac%7B5%7D%7B9%7D%2C%20%5Cfrac%7B10%7D%7B27%7D%2C%20%5Cfrac%7B20%7D%7B81%7D%2C%20%5Cldots)
To find the 5th term of the given geometric sequence:
Let
etc.
First find the common ratio
![r=\frac{a_2}{a_1}](https://tex.z-dn.net/?f=r%3D%5Cfrac%7Ba_2%7D%7Ba_1%7D)
![=\frac{\frac{5}{9}}{\frac{5}{6}}](https://tex.z-dn.net/?f=%3D%5Cfrac%7B%5Cfrac%7B5%7D%7B9%7D%7D%7B%5Cfrac%7B5%7D%7B6%7D%7D)
![=\frac{5}{9}\times \frac{6}{5}](https://tex.z-dn.net/?f=%3D%5Cfrac%7B5%7D%7B9%7D%5Ctimes%20%5Cfrac%7B6%7D%7B5%7D)
Therefore ![r=\frac{2}{3}](https://tex.z-dn.net/?f=r%3D%5Cfrac%7B2%7D%7B3%7D)
![r=\frac{a_3}{a_2}](https://tex.z-dn.net/?f=r%3D%5Cfrac%7Ba_3%7D%7Ba_2%7D)
![=\frac{\frac{10}{27}}{\frac{5}{9}}](https://tex.z-dn.net/?f=%3D%5Cfrac%7B%5Cfrac%7B10%7D%7B27%7D%7D%7B%5Cfrac%7B5%7D%7B9%7D%7D)
![=\frac{10}{27}\times \frac{9}{5}](https://tex.z-dn.net/?f=%3D%5Cfrac%7B10%7D%7B27%7D%5Ctimes%20%5Cfrac%7B9%7D%7B5%7D)
Therefore ![r=\frac{2}{3}](https://tex.z-dn.net/?f=r%3D%5Cfrac%7B2%7D%7B3%7D)
Therefore the common ratio ![r=\frac{2}{3}](https://tex.z-dn.net/?f=r%3D%5Cfrac%7B2%7D%7B3%7D)
The nth term of geometric sequence is ![a_n=a_1r^{n-1}](https://tex.z-dn.net/?f=a_n%3Da_1r%5E%7Bn-1%7D)
Put
and
in the above equation we get
![a_5=\frac{5}{6}(\frac{2}{3})^{5-1}](https://tex.z-dn.net/?f=a_5%3D%5Cfrac%7B5%7D%7B6%7D%28%5Cfrac%7B2%7D%7B3%7D%29%5E%7B5-1%7D)
![=\frac{5}{6}(\frac{2}{3})^4](https://tex.z-dn.net/?f=%3D%5Cfrac%7B5%7D%7B6%7D%28%5Cfrac%7B2%7D%7B3%7D%29%5E4)
![=\frac{5}{6}(\frac{2}{3}\times \frac{2}{3}\times \frac{2}{3}\times \frac{2}{3})](https://tex.z-dn.net/?f=%3D%5Cfrac%7B5%7D%7B6%7D%28%5Cfrac%7B2%7D%7B3%7D%5Ctimes%20%5Cfrac%7B2%7D%7B3%7D%5Ctimes%20%5Cfrac%7B2%7D%7B3%7D%5Ctimes%20%5Cfrac%7B2%7D%7B3%7D%29)
![=\frac{5}{3}(\frac{8}{81})](https://tex.z-dn.net/?f=%3D%5Cfrac%7B5%7D%7B3%7D%28%5Cfrac%7B8%7D%7B81%7D%29)
![a_5=\frac{40}{243}](https://tex.z-dn.net/?f=a_5%3D%5Cfrac%7B40%7D%7B243%7D)
Therefore ![a_5=\frac{40}{243}](https://tex.z-dn.net/?f=a_5%3D%5Cfrac%7B40%7D%7B243%7D)
Therefore the next fraction in the given geometric sequence is ![\frac{40}{243}](https://tex.z-dn.net/?f=%5Cfrac%7B40%7D%7B243%7D)
Lin's number is 15 and Jen's number is 105.
15 is a two digit number and if we multiply it with 7, we get the answer 105.
15 x 7 = 105
And if we put 0 between the Lin's number that is 15, we will get 105
Thus, 15 and 105 are two numbers.
Answer : y = -2x - 4
All you have to do is put y on one side and the 2x on the other side, just make sure to make it negative since you are subtracting it to go on the other side
4x + 2y = -16
2x + y = - 8 (divide both side by 2)
so
y = -2x - 8
answer
y = -2x - 8