Answer:
4/(3(x-2))
Step-by-step explanation:
3x^2-21x+30=3(x^2-7x+10)=3(x-5)(x-2)
3x-15=3(x-5)
----------------------------
So the common denominator must be 3(x-5)(x-2)
2(x-2)=2x-4
Add the numerators,
(2x-16)+(2x-4)=4x-16-4=4x-20
-----------------
(4x-20)/[3(x-5)(x-2)]
simplify 4x-20 into 4(x-5)
cancel out the (x-5)'s for both the denominator and the numerator
4/[3(x-2)]
The answer is the top right one.
Each of the other answers is saying the same thing, just in different words, that the slope of the equation is 2. The top right is incorrect.
Answer:

Given expression is
![\rm :\longmapsto\:\displaystyle\lim_{n \to \infty }\rm \bigg[\dfrac{1}{3} + \dfrac{1}{ {3}^{2} } + \dfrac{1}{ {3}^{3} } + - - + \dfrac{1}{ {3}^{n} } \bigg]](https://tex.z-dn.net/?f=%5Crm%20%3A%5Clongmapsto%5C%3A%5Cdisplaystyle%5Clim_%7Bn%20%5Cto%20%20%5Cinfty%20%7D%5Crm%20%5Cbigg%5B%5Cdfrac%7B1%7D%7B3%7D%20%2B%20%5Cdfrac%7B1%7D%7B%20%7B3%7D%5E%7B2%7D%20%7D%20%20%2B%20%5Cdfrac%7B1%7D%7B%20%7B3%7D%5E%7B3%7D%20%7D%20%20%2B%20%20-%20%20-%20%20%2B%20%5Cdfrac%7B1%7D%7B%20%7B3%7D%5E%7Bn%7D%20%7D%20%20%5Cbigg%5D)
Let we first evaluate

Its a Geometric progression with



So, Sum of n terms of GP series is

![\rm :\longmapsto\:S_n = \dfrac{1}{3} \bigg[\dfrac{1 - {\bigg[\dfrac{1}{3} \bigg]}^{n} }{1 - \dfrac{1}{3} } \bigg]](https://tex.z-dn.net/?f=%5Crm%20%3A%5Clongmapsto%5C%3AS_n%20%3D%20%5Cdfrac%7B1%7D%7B3%7D%20%5Cbigg%5B%5Cdfrac%7B1%20-%20%20%7B%5Cbigg%5B%5Cdfrac%7B1%7D%7B3%7D%20%5Cbigg%5D%7D%5E%7Bn%7D%20%7D%7B1%20-%20%5Cdfrac%7B1%7D%7B3%7D%20%7D%20%5Cbigg%5D)
![\rm :\longmapsto\:S_n = \dfrac{1}{3} \bigg[\dfrac{1 - {\bigg[\dfrac{1}{3} \bigg]}^{n} }{\dfrac{3 - 1}{3} } \bigg]](https://tex.z-dn.net/?f=%5Crm%20%3A%5Clongmapsto%5C%3AS_n%20%3D%20%5Cdfrac%7B1%7D%7B3%7D%20%5Cbigg%5B%5Cdfrac%7B1%20-%20%20%7B%5Cbigg%5B%5Cdfrac%7B1%7D%7B3%7D%20%5Cbigg%5D%7D%5E%7Bn%7D%20%7D%7B%5Cdfrac%7B3%20-%201%7D%7B3%7D%20%7D%20%5Cbigg%5D)
![\rm :\longmapsto\:S_n = \dfrac{1}{3} \bigg[\dfrac{1 - {\bigg[\dfrac{1}{3} \bigg]}^{n} }{\dfrac{2}{3} } \bigg]](https://tex.z-dn.net/?f=%5Crm%20%3A%5Clongmapsto%5C%3AS_n%20%3D%20%5Cdfrac%7B1%7D%7B3%7D%20%5Cbigg%5B%5Cdfrac%7B1%20-%20%20%7B%5Cbigg%5B%5Cdfrac%7B1%7D%7B3%7D%20%5Cbigg%5D%7D%5E%7Bn%7D%20%7D%7B%5Cdfrac%7B2%7D%7B3%7D%20%7D%20%5Cbigg%5D)
![\bf\implies \:S_n = \dfrac{1}{2}\bigg[1 - \dfrac{1}{ {3}^{n} } \bigg]](https://tex.z-dn.net/?f=%5Cbf%5Cimplies%20%5C%3AS_n%20%3D%20%5Cdfrac%7B1%7D%7B2%7D%5Cbigg%5B1%20-%20%5Cdfrac%7B1%7D%7B%20%7B3%7D%5E%7Bn%7D%20%7D%20%5Cbigg%5D)
<u>Hence, </u>
![\bf :\longmapsto\:\dfrac{1}{3} + \dfrac{1}{ {3}^{2} } + \dfrac{1}{ {3}^{3} } + - - + \dfrac{1}{ {3}^{n} } = \dfrac{1}{2}\bigg[1 - \dfrac{1}{ {3}^{n} } \bigg]](https://tex.z-dn.net/?f=%5Cbf%20%3A%5Clongmapsto%5C%3A%5Cdfrac%7B1%7D%7B3%7D%20%2B%20%5Cdfrac%7B1%7D%7B%20%7B3%7D%5E%7B2%7D%20%7D%20%20%2B%20%5Cdfrac%7B1%7D%7B%20%7B3%7D%5E%7B3%7D%20%7D%20%20%2B%20%20-%20%20-%20%20%2B%20%5Cdfrac%7B1%7D%7B%20%7B3%7D%5E%7Bn%7D%20%7D%20%3D%20%5Cdfrac%7B1%7D%7B2%7D%5Cbigg%5B1%20-%20%5Cdfrac%7B1%7D%7B%20%7B3%7D%5E%7Bn%7D%20%7D%20%5Cbigg%5D)
<u>Therefore, </u>
![\purple{\rm :\longmapsto\:\displaystyle\lim_{n \to \infty }\rm \bigg[\dfrac{1}{3} + \dfrac{1}{ {3}^{2} } + \dfrac{1}{ {3}^{3} } + - - + \dfrac{1}{ {3}^{n} } \bigg]}](https://tex.z-dn.net/?f=%20%5Cpurple%7B%5Crm%20%3A%5Clongmapsto%5C%3A%5Cdisplaystyle%5Clim_%7Bn%20%5Cto%20%20%5Cinfty%20%7D%5Crm%20%5Cbigg%5B%5Cdfrac%7B1%7D%7B3%7D%20%2B%20%5Cdfrac%7B1%7D%7B%20%7B3%7D%5E%7B2%7D%20%7D%20%20%2B%20%5Cdfrac%7B1%7D%7B%20%7B3%7D%5E%7B3%7D%20%7D%20%20%2B%20%20-%20%20-%20%20%2B%20%5Cdfrac%7B1%7D%7B%20%7B3%7D%5E%7Bn%7D%20%7D%20%20%5Cbigg%5D%7D)
![\rm \: = \: \displaystyle\lim_{n \to \infty }\rm \dfrac{1}{2}\bigg[1 - \dfrac{1}{ {3}^{n} } \bigg]](https://tex.z-dn.net/?f=%5Crm%20%5C%3A%20%20%3D%20%20%5C%3A%20%5Cdisplaystyle%5Clim_%7Bn%20%5Cto%20%20%5Cinfty%20%7D%5Crm%20%5Cdfrac%7B1%7D%7B2%7D%5Cbigg%5B1%20-%20%5Cdfrac%7B1%7D%7B%20%7B3%7D%5E%7Bn%7D%20%7D%20%5Cbigg%5D)
![\rm \: = \: \rm \dfrac{1}{2}\bigg[1 - 0 \bigg]](https://tex.z-dn.net/?f=%5Crm%20%5C%3A%20%20%3D%20%20%5C%3A%20%5Crm%20%5Cdfrac%7B1%7D%7B2%7D%5Cbigg%5B1%20-%200%20%5Cbigg%5D)

<u>Hence, </u>
![\purple{\rm :\longmapsto\:\boxed{\tt{ \displaystyle\lim_{n \to \infty }\rm \bigg[\dfrac{1}{3} + \dfrac{1}{ {3}^{2} } + \dfrac{1}{ {3}^{3} } + - - + \dfrac{1}{ {3}^{n} } \bigg]} = \frac{1}{2}}}](https://tex.z-dn.net/?f=%20%5Cpurple%7B%5Crm%20%3A%5Clongmapsto%5C%3A%5Cboxed%7B%5Ctt%7B%20%5Cdisplaystyle%5Clim_%7Bn%20%5Cto%20%20%5Cinfty%20%7D%5Crm%20%5Cbigg%5B%5Cdfrac%7B1%7D%7B3%7D%20%2B%20%5Cdfrac%7B1%7D%7B%20%7B3%7D%5E%7B2%7D%20%7D%20%20%2B%20%5Cdfrac%7B1%7D%7B%20%7B3%7D%5E%7B3%7D%20%7D%20%20%2B%20%20-%20%20-%20%20%2B%20%5Cdfrac%7B1%7D%7B%20%7B3%7D%5E%7Bn%7D%20%7D%20%20%5Cbigg%5D%7D%20%3D%20%20%5Cfrac%7B1%7D%7B2%7D%7D%7D)
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<h3>
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