If `a_(a), a _(2), a _(3),…., a_(n)` are in H.P. and `f (k)=sum _(r =1) ^(n) a_(r)-a_(k) ` then `(a_(1))/(f(1)), (a_(2))/(f (2)), (a_(3))/(f (n))` are in :

If a_(1),a_(2),".....", are in HP and f_(k)=sum_(r=1)^(n)a_(r)-a_(k) , then 2^(alpha_(1)),2^(alpha_(2)),2^(alpha_(3))2^(alpha_(4)),"....." are in {" where " alpha_(1)=(a_(1))/(f_(1)),alpha_(2)=(a_(2))/(f_(2)),alpha_(3)=(a_(3))/(f_(3)),"....."} .

If a_(1),a_(2)a_(3),….,a_(15) are in A.P and a_(1)+a_(8)+a_(15)=15 , then a_(2)+a_(3)+a_(8)+a_(13)+a_(14) is equal to

Let (1 + x)^(n) = sum_(r=0)^(n) a_(r) x^(r) . Then ( a+ (a_(1))/(a_(0))) (1 + (a_(2))/(a_(1)))…(1 + (a_(n))/(a_(n-1))) is equal to

If a_(1),a_(2),a_(3),a_(4),a_(5) are in HP, then a_(1)a_(2)+a_(2)a_(3)+a_(3)a_(4)+a_(4)a_(5) is equal to

if a_(1),a_(2),a_(3)……,a_(12) are in A.P and Delta_(1)= |{:(a_(1)a_(5),,a_(1),,a_(2)),(a_(2)a_(6),,a_(2),,a_(3)),(a_(3)a_(7),,a_(3),,a_(4)):}| Delta_(2)= |{:(a_(2)a_(10),,a_(2),,a_(3)),(a_(3)a_(11),,a_(3),,a_(4)),(a_(4)a_(12),,a_(4),,a_(5)):}| then Delta_(1):Delta_(2)= "_____"

If a_(1),a_(2),a_(3),".....",a_(n) are in HP, than prove that a_(1)a_(2)+a_(2)a_(3)+a_(3)a_(4)+"....."+a_(n-1)a_(n)=(n-1)a_(1)a_(n)

If a_(1),a_(2),a_(3),"........",a_(n) are in AP with a_(1)=0 , prove that (a_(3))/(a_(2))+(a_(4))/(a_(3))+"......"+(a_(n))/(a_(n-1))-a_(2)((1)/(a_(2))+(1)/(a_(3))"+........"+(1)/(a_(n-2)))=(a_(n-1))/(a_(2))+(a_(2))/(a_(n-1)) .

If a_(1)=2 and a_(n)=2a_(n-1)+5 for ngt1 , the value of sum_(r=2)^(5)a_(r) is

If A_(1), A_(2),..,A_(n) are any n events, then

Let a_(1),a_(2),a_(3), . . .,a_(n) be an A.P. Statement -1 : (1)/(a_(1)a_(n))+(1)/(a_(2)a_(n-1))+(1)/(a_(3)a_(n-1))+ . . .. +(1)/(a_(n)a_(1)) =(2)/(a_(1)+a_(n))((1)/(a_(1))+(1)/(a_(2))+ . . .. +(1)/(a_(n))) Statement -2: a_(r)+a_(n-r+1)=a_(1)+a_(n)" for "1lerlen