If `a_1,a_2,a_3,...` are in A.P. and `a_i>0` for each i, then `sum_(i=1)^n n/(a_(i+1)^(2/3)+a_(i+1)^(1/3)a_i^(1/3)+a_i^(2/3))` is equal to

If a_1,a_2,a_3,... are in A.P. and a_i>0 for each i, then sum_(i=1)^n n/(a_(i+1)^(2/3)+a_(i+1)^(1/3)a_i^(1/3)+a_i^(2/3)) is equal to (a) (n)/(a_(n)^(2//3)+a_(n)^(1//3)+a_(1)^(2//3)) (b) (n(n+1))/(a_(n)^(2//3)+a_(n)^(1//3)+a_(1)^(2//3)) (c) (n(n-1))/(a_(n)^(2//3)+a_(n)^(1//3)*a_(1)^(1//3)+a_(1)^(2//3)) (d) None of these

If a_(1),a_(2),a_(3), . .. . ,a_(n) are in A.P. and a_(i)gt0 for each i=1,2,3, . . ..,n, then sum_(r=1)^(n-1)(1)/(a_(r+1)^(2//3)+a_(r+1)^(2//3)a_(r)^(1//3)+a_(r)^(1//3)) is equal to

If a_1,a_2,a_3,...,a_n be in AP whose common difference is d then prove that sum_(i=1)^n a_ia_(i+1)=n{a_1^2+na_1d+(n^2-1)/3 d^2} .

If a_1,a_2,a_3,….,a_n are in G.P. are in a_i > 0 for each I, then the determinant Delta=|{:(loga_n,log a_(n+2),log a_(n+4)),(log a_(n+6),log a_(n+8), log a_(n+10)),(log a_(n+12), log a_(n+14), loga_(n+16)):}|

If a_(1) , a_(2) , a_(3),"…………."a_(n) are in A.P., where a_(i) gt 0 for all i, then the value of : (1) /( sqrt(a_(1))+sqrt(a_(2)))+ (1) /( sqrt(a_(2))+sqrt(a_(3)))+"......"+(1) /( sqrt(a_(n-1))+sqrt(a_(n))) is :

a_1,a_2,a_3 ,…., a_n from an A.P. Then the sum sum_(i=1)^10 (a_i a_(i+1)a_(i+2))/(a_i + a_(i+2)) where a_1=1 and a_2=2 is :

if a,A_(1),A_(2),A_(3)......,A_(2n),b are in A.P.then sum_(i=1)^(2n)A_(i)=(a+b)