If `a_1,a_2...................,a_19` are the first `19` term of an `AP and a_1+a_8+a_12+a_19=224.` Then `sum_(i=1)^19 a_i` is equal to :

a_1, a_2, a_3 …..a_9 are in GP where a_1 lt 0, a_1 + a_2 = 4, a_3 + a_4 = 16 , if sum_(i=1)^9 a_i = 4 lambda then lambda is equal to

Let a_1,a_2,a_3,.... are in GP. If a_n>a_m when n>m and a_1+a_n=66 while a_2*a_(n-1)=128 and sum_(i=1)^n a_i=126 , find the value of n .

A_1 A_2 A_3............A_n is a polygon whose all sides are not equal, and angles too are not all equal. theta_1 is theangle between bar(A_i A_j +1) and bar(A_(i+1) A_(i +2)),1 leq i leq n-2.theta_(n-1) is the angle between bar(A_(n-1) A_n) and bar(A_n A_1), while theta_n, is the angle between bar(A_n A_1) and bar(A_1 A_2) then cos(sum_(i=1)^n theta_i) is equal to-

Find the indicated term of each G.P. a_1 = 8, r =1/2 , find a_8

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

Let A={a_1,a_2,a_3,.....} and B={b_1,b_2,b_3,.....} are arithmetic sequences such that a_1=b_1!=0, a_2=2b_2 and sum_(i=1)^10 a_i=sum_(i=1)^15 b_j , If (a_2-a_1)/(b_2-b_1)=p/q where p and q are coprime then find the value of (p+q).

Let a_1 , a_2 , a_3,.... be terms of an A.P. if (a_1 + a_2 +....+ a_p)/(a_1 + a_2 +...+ a_q) = p^2/q^2. p ne q then a_6/a_21 equals

If a_1, a_2,a_3,..........., a_n be an A.P. of non-zero terms, prove that : 1/(a_1 a_2)+1/(a_2 a_3)+ ............. + 1/(a_(n-1) a_n)= (n-1)/(a_1 a_n) .

If a_1, a_2 , ............, a_n are in A.P. and a_i >0 for all i , prove that : 1/(sqrt a_1+sqrt a_2)+1/(sqrt a_2+sqrt a_3)+.............+ 1/(sqrt (a_(n-1))+sqrt a_n)= (n-1)/(sqrt a_1+sqrt a_n) .

If a_1+a_2+a_3+......+a_n=1 AA a_i > 0, i=1,2,3,......,n , then find the maximum value of a_1 a_2 a_3 a_4 a_5......a_n .