Match the following Column I to Column II

Explanation

(A) `a_(1),a_(2),a_(3),"......."a_(n),"......."` are in AP
and `a_(1)=(5)/(2),a_(10)=16`
`:.a_(1)+a_(2)+"...."a_(n)=110`
`implies (n)/(2)(a_(1)+a_(n))=110`
`implies (n)/(2)[(5)/(2)+(5)/(2)+(n-1)d]=110" " ".......(i)"`
Now, `d=(a_(10)-a_(1))/(10-1)=(16-(5)/(2))/(9)=(27)/(9xx2)=(3)/(2)" " "......(ii)"`
From Eqs. (i) and (ii), we get `(n)/(2)[5+(n-1)(3)/(2)]=110`
`implies 5n+(n^(2)-n)(3)/(2)=220`
`implies3n^(2)+7n-440=0`
`implies 3n^(2)+40n-33n-440=0`
`implies n(3n+40)-11(3n+40)=0`
`implies (3n+40)(n-11)=0`
So, `n=-(40)/(3)" or "n=11`
`:. n=11 " " " " [ n in N]`.
(B) Let first angle` =a " " " " [" in degrees "]`
Common difference `=d " " " " [" in degrees "]`
Number of sides n=9
`:.` Sum of interior angles `=(n-2)xx180^(@)`
`implies(n)/(2)[2a+(n-1)d]=(n-2)xx180^(@)`
`implies (9)/(2)(2a+8d)=7xx180^(@)`
`implies a+4d=140^(@)`
and largest angle `T_(9)=a+8dlt180^(@)`
`implies 4dlt40`
`implies dlt10`
`:.d=9`
(C ) Given increasing GP,
`a_(1),a_(2),"....,"a_(n)"...."`
where `a_(6)=4aa_(4)`
`a_(1)r(^5) =4a_(1)r^(3) " " [" r is the common ratio "]`
`implies r^(2)=4`
`implies r=2 " " [:. " increasing GP "]`
and `a_(9)-a_(7) =192`
`a_(1)(r^(8)-r^(6))=192`
`a_(1)(256-64)=192`
`a_(1)=(192)/(192)`
`a_(1)=1`
Then, `a_(2)=2,a_(3)=4 " and "a_(4)+a_(5)+"...."a_(n)=1016`
`(a_(1)+a_(2)+"......"+a_(n))-(a_(1)+a_(2)+a_(3))=1016`
`(1(2^(n)))/(2-1)=1016+7`
`2^(n)=1016+1=1024=2^(10)`
`:. n=10`.