Match the following Column I to Column II

Answer

`(A)to(q);(B)to(p);(C )to(s)`

Explanation

(A)`:.a_(1),a_(2),a_(3),"...."` are in AP.
`:.a_(1)+a_(30)=a_(6)+a_(25)=a_(10)+a_(21)=lambda`
`:.a_(1)+a_(6)+a_(10)+a_(21)+a_(25)+a_(30)=120`
`:.3 lambda=120`
`implies lambda=40`
Then, `sum_(i=1)^(30)a_(i)=(30)/(2)(a_(1)+a_(30))=15xxlambda=15xx40=600`
(B) `a_(1),a_(2),a_(3),"...."` are in AP.
`:.a_(1)+a_(25)=a_(5)a_(21)`
`=a_(9)+a_(17)=a_(13)+a_(13)=lambda [" say "]`
`:.=a_(1)+a_(5)+a_(9)+a_(13)+a_(17)+a_(21)+a_(25)=112`
`:.3lambda+(lambda)/(2)=112`
`implies (7lambda)/(2)=112`
`implies lambda=32`
Then, `sum_(i=1)^(25)a_(i)=(25)/(2)(a_(1)+a_(25))=(25)/(2)xx32=400`
(C ) `:.a_(1),a_(2),a_(3)."...."` are in AP.
`:.a_(1)+a_(16)=a_(4)+a_(13)=a_(7)+a_(10)lambda" " [" say "]`
`:.a_(1)+a_(4)+a_(7)+a_(10)+a_(13)+a_(16)=375`
`:.3lambda=375 :.lambda=125`
`sum_(i=1)^(16)a_(i)=(16)/(2)(a_(1)+a_(16))=8xxlambda=8xx125=1000`