![\underline{\underline{\large\bf{Given:-}}}](https://tex.z-dn.net/?f=%5Cunderline%7B%5Cunderline%7B%5Clarge%5Cbf%7BGiven%3A-%7D%7D%7D)
![\red{\leadsto}\:](https://tex.z-dn.net/?f=%5Cred%7B%5Cleadsto%7D%5C%3A)
![\textsf{}](https://tex.z-dn.net/?f=%5Ctextsf%7B%7D%20)
![\sf Number \: of \:terms \: in \: A.P,n = 30](https://tex.z-dn.net/?f=%5Csf%20Number%20%5C%3A%20of%20%20%5C%3Aterms%20%5C%3A%20in%20%5C%3A%20A.P%2Cn%20%3D%2030)
![\red{\leadsto}\:](https://tex.z-dn.net/?f=%5Cred%7B%5Cleadsto%7D%5C%3A)
![\textsf{}](https://tex.z-dn.net/?f=%5Ctextsf%7B%7D%20)
![\sf Fourth \: term ,a_4 = 11](https://tex.z-dn.net/?f=%5Csf%20Fourth%20%5C%3A%20term%20%2Ca_4%20%3D%2011%20%20)
![\red{\leadsto}\:](https://tex.z-dn.net/?f=%5Cred%7B%5Cleadsto%7D%5C%3A)
![\textsf{}](https://tex.z-dn.net/?f=%5Ctextsf%7B%7D%20)
![\sf last\:term, a_{30} = 89](https://tex.z-dn.net/?f=%5Csf%20last%5C%3Aterm%2C%20a_%7B30%7D%20%3D%2089%20%20)
![\underline{\underline{\large\bf{To Find:-}}}](https://tex.z-dn.net/?f=%5Cunderline%7B%5Cunderline%7B%5Clarge%5Cbf%7BTo%20Find%3A-%7D%7D%7D)
![\orange{\leadsto}\:](https://tex.z-dn.net/?f=%5Corange%7B%5Cleadsto%7D%5C%3A)
![\textsf{ }](https://tex.z-dn.net/?f=%5Ctextsf%7B%20%7D%20)
![\sf The \: A.P.](https://tex.z-dn.net/?f=%5Csf%20The%20%5C%3A%20A.P.%20)
![\orange{\leadsto}\:](https://tex.z-dn.net/?f=%5Corange%7B%5Cleadsto%7D%5C%3A)
![\textsf{ }](https://tex.z-dn.net/?f=%5Ctextsf%7B%20%7D%20)
![\sf 23rd\: term, a_{23}](https://tex.z-dn.net/?f=%5Csf%2023rd%5C%3A%20term%2C%20a_%7B23%7D%20)
![\\](https://tex.z-dn.net/?f=%5C%5C)
![\underline{\underline{\large\bf{Solution:-}}}\\](https://tex.z-dn.net/?f=%5Cunderline%7B%5Cunderline%7B%5Clarge%5Cbf%7BSolution%3A-%7D%7D%7D%5C%5C)
The nth term of A.P is determined by the formula-
![\green{ \underline { \boxed{ \sf{a_n = a+(n-1)d}}}}](https://tex.z-dn.net/?f=%5Cgreen%7B%20%5Cunderline%20%7B%20%5Cboxed%7B%20%5Csf%7Ba_n%20%3D%20a%2B%28n-1%29d%7D%7D%7D%7D)
where
Since ,
![\sf a_4 = 11](https://tex.z-dn.net/?f=%5Csf%20a_4%20%3D%2011%20%20)
![\sf a+(4-1) d= 11](https://tex.z-dn.net/?f=%5Csf%20a%2B%284-1%29%20d%3D%2011%20%20)
![\sf a+3d= 11\_\_\_(1)](https://tex.z-dn.net/?f=%5Csf%20a%2B3d%3D%2011%5C_%5C_%5C_%281%29%20%20)
![\sf a_{30}= 89](https://tex.z-dn.net/?f=%5Csf%20a_%7B30%7D%3D%2089%20)
![\sf a+(30-1)d=89](https://tex.z-dn.net/?f=%5Csf%20a%2B%2830-1%29d%3D89)
![\longrightarrow](https://tex.z-dn.net/?f=%5Clongrightarrow)
![\sf a+29d= 89\_\_\_(2)](https://tex.z-dn.net/?f=%5Csf%20a%2B29d%3D%2089%5C_%5C_%5C_%282%29)
<u>Subtracting equation (1) from equation(2)</u>
![\begin{gathered}\\\implies\quad \sf a+29d-(a+3d) = 89-11 \\\end{gathered}](https://tex.z-dn.net/?f=%5Cbegin%7Bgathered%7D%5C%5C%5Cimplies%5Cquad%20%5Csf%20a%2B29d-%28a%2B3d%29%20%3D%2089-11%20%5C%5C%5Cend%7Bgathered%7D%20)
![\begin{gathered}\\\implies\quad \sf a+29d-a-3d = 78 \\\end{gathered}](https://tex.z-dn.net/?f=%5Cbegin%7Bgathered%7D%5C%5C%5Cimplies%5Cquad%20%5Csf%20a%2B29d-a-3d%20%3D%2078%20%5C%5C%5Cend%7Bgathered%7D%20)
![\begin{gathered}\\\implies\quad \sf a-a+29d-3d = 78 \\\end{gathered}](https://tex.z-dn.net/?f=%5Cbegin%7Bgathered%7D%5C%5C%5Cimplies%5Cquad%20%5Csf%20a-a%2B29d-3d%20%3D%2078%20%5C%5C%5Cend%7Bgathered%7D%20)
![\begin{gathered}\\\implies\quad \sf 26d = 78 \\\end{gathered}](https://tex.z-dn.net/?f=%5Cbegin%7Bgathered%7D%5C%5C%5Cimplies%5Cquad%20%5Csf%2026d%20%3D%2078%20%5C%5C%5Cend%7Bgathered%7D%20)
![\begin{gathered}\\\implies\quad \sf d = \frac{78}{26} \\\end{gathered}](https://tex.z-dn.net/?f=%5Cbegin%7Bgathered%7D%5C%5C%5Cimplies%5Cquad%20%5Csf%20d%20%3D%20%5Cfrac%7B78%7D%7B26%7D%20%5C%5C%5Cend%7Bgathered%7D%20)
![\begin{gathered}\\\implies\quad \sf d = 3 \\\end{gathered}](https://tex.z-dn.net/?f=%5Cbegin%7Bgathered%7D%5C%5C%5Cimplies%5Cquad%20%5Csf%20d%20%3D%203%20%5C%5C%5Cend%7Bgathered%7D%20)
Putting the value of d in equation (1) -
![\begin{gathered}\\\implies\quad \sf a+3(3) = 11 \\\end{gathered}](https://tex.z-dn.net/?f=%5Cbegin%7Bgathered%7D%5C%5C%5Cimplies%5Cquad%20%5Csf%20a%2B3%283%29%20%3D%2011%20%5C%5C%5Cend%7Bgathered%7D%20)
![\begin{gathered}\\\implies\quad \sf a = 11-9 \\\end{gathered}](https://tex.z-dn.net/?f=%5Cbegin%7Bgathered%7D%5C%5C%5Cimplies%5Cquad%20%5Csf%20a%20%3D%2011-9%20%5C%5C%5Cend%7Bgathered%7D%20)
![\begin{gathered}\\\implies\quad \sf a = 2 \\\end{gathered}](https://tex.z-dn.net/?f=%5Cbegin%7Bgathered%7D%5C%5C%5Cimplies%5Cquad%20%5Csf%20a%20%3D%202%20%5C%5C%5Cend%7Bgathered%7D%20)
- Second term of A.P.,
![\quad\quad\quad\sf =2+3](https://tex.z-dn.net/?f=%5Cquad%5Cquad%5Cquad%5Csf%20%3D2%2B3)
![\quad\quad\quad\sf =5](https://tex.z-dn.net/?f=%5Cquad%5Cquad%5Cquad%5Csf%20%3D5%20)
- Third term of A.P.,
![\quad\quad\quad\sf =2+6](https://tex.z-dn.net/?f=%5Cquad%5Cquad%5Cquad%5Csf%20%3D2%2B6)
![\quad\quad\quad\sf =8](https://tex.z-dn.net/?f=%5Cquad%5Cquad%5Cquad%5Csf%20%3D8%20)
Thus , The A.P is 2,5,8,. . . . . .
<u>Now,</u>
![\begin{gathered}\\\implies\quad \sf a_n = a+(n-1)d \\\end{gathered}](https://tex.z-dn.net/?f=%5Cbegin%7Bgathered%7D%5C%5C%5Cimplies%5Cquad%20%5Csf%20a_n%20%3D%20a%2B%28n-1%29d%20%5C%5C%5Cend%7Bgathered%7D%20)
![\begin{gathered}\\\implies\quad \sf a_{23 }= 2+(23-1)\times 3 \\\end{gathered}](https://tex.z-dn.net/?f=%5Cbegin%7Bgathered%7D%5C%5C%5Cimplies%5Cquad%20%5Csf%20a_%7B23%20%7D%3D%202%2B%2823-1%29%5Ctimes%203%20%5C%5C%5Cend%7Bgathered%7D%20)
![\begin{gathered}\\\implies\quad \sf a_{23} = 2+22 \times 3 \\\end{gathered}](https://tex.z-dn.net/?f=%5Cbegin%7Bgathered%7D%5C%5C%5Cimplies%5Cquad%20%5Csf%20a_%7B23%7D%20%3D%202%2B22%20%5Ctimes%203%20%20%5C%5C%5Cend%7Bgathered%7D%20)
![\begin{gathered}\\\implies\quad \sf a_{23} = 2+66 \\\end{gathered}](https://tex.z-dn.net/?f=%5Cbegin%7Bgathered%7D%5C%5C%5Cimplies%5Cquad%20%5Csf%20a_%7B23%7D%20%3D%202%2B66%20%20%5C%5C%5Cend%7Bgathered%7D%20)
![\begin{gathered}\\\implies\quad \sf a_{23} = 68 \\\end{gathered}](https://tex.z-dn.net/?f=%5Cbegin%7Bgathered%7D%5C%5C%5Cimplies%5Cquad%20%5Csf%20a_%7B23%7D%20%3D%2068%20%5C%5C%5Cend%7Bgathered%7D%20)
Thus , 23rd term is 68.
180 minus 47 is 133 so b would be the answer because it’s a straight line which is automatically 180
If you do the math it should be -7.78725563766 which is rounded to -8
Answer:
<em>It can also be expressed as 35/14.</em>
Step-by-step explanation:
<u>Equivalent Ratio</u>
A ratio is a relation between two numbers a, b in the format a:b and it's equivalent to the division a/b.
Since it's equivalent to a division or a fraction, it can be simplified or amplified by multiplying or dividing both sides by the same number.
The ratio 5:2 can be amplified by multiplied by 7 as follows:
5*7:2*7 = 35:14.
It can also be expressed as 35/14.