[" Let "a_(1),a_(2),a_(3),...,a_(10)" be in G.P.with "a_(i)>0" for "i=1,2,...,10" and "S" be the set of pairs "(r,k)," r "k in N],[" (the set of natural numbers) for which "log_(e)a_(i)^(r)(k)/(q_(i))log_(a)a_(j)^(r)a_(b)^(k)log_(e)a_(i)^(r)a_(i)^(r)|],[" (the set of natural numbers) for which "log_(e)a_(i)^(r)a_(b)^(k)log_(a)a_(b)^(i)a_(b)^(k)log_(c)a_(s)^(i)a_(j)^(k)a_(i0)^(k)],[" Then the number of elements in "S" ,is: "]

Let a_(1),a_(2),a_(3), …, a_(10) be in G.P. with a_(i) gt 0 for i=1, 2, …, 10 and S be te set of pairs (r, k), r, k in N (the set of natural numbers) for which |(log_(e)a_(1)^(r)a_(2)^(k),log_(e)a_(2)^(r)a_(3)^(k),log_(e)a_(3)^(r)a_(4)^(k)),(log_(e)a_(4)^(r)a_(5)^(k),log_(e)a_(5)^(r)a_(6)^(k),log_(e)a_(6)^(r)a_(7)^(k)),(log_(e)a_(7)^(r)a_(8)^(k),log_(e)a_(8)^(r)a_(9)^(k),log_(e)a_(9)^(r)a_(10)^(k))| = 0. Then the number of elements in S is

Let a_(1),a_(2),a_(3), …, a_(10) be in G.P. with a_(i) gt 0 for i=1, 2, …, 10 and S be te set of pairs (r, k), r, k in N (the set of natural numbers) for which |(log_(e)a_(1)^(r)a_(2)^(k),log_(e)a_(2)^(r)a_(3)^(k),log_(e)a_(3)^(r)a_(4)^(k)),(log_(e)a_(4)^(r)a_(5)^(k),log_(e)a_(5)^(r)a_(6)^(k),log_(e)a_(6)^(r)a_(7)^(k)),(log_(e)a_(7)^(r)a_(8)^(k),log_(e)a_(8)^(r)a_(9)^(k),log_(e)a_(9)^(r)a_(10)^(k))| = 0. Then the number of elements in S is

Let a_(1),a_(2),a_(3), …, a_(10) be in G.P. with a_(i) gt 0 for i=1, 2, …, 10 and S be te set of pairs (r, k), r, k in N (the set of natural numbers) for which |(log_(e)a_(1)^(r)a_(2)^(k),log_(e)a_(2)^(r)a_(3)^(k),log_(e)a_(3)^(r)a_(4)^(k)),(log_(e)a_(4)^(r)a_(5)^(k),log_(e)a_(5)^(r)a_(6)^(k),log_(e)a_(6)^(r)a_(7)^(k)),(log_(e)a_(7)^(r)a_(8)^(k),log_(e)a_(8)^(r)a_(9)^(k),log_(e)a_(9)^(r)a_(10)^(k))| = 0. Then the number of elements in S is

Let a_(1), a_(2), a_(3)……, a_(10) " be in GP with " a_(i) gt " for " I = 2,2,…..,10 and S be the set of pairs (r,k), r, k in N (the set of natural numbers) for which [{:("log"_(e)a_(1)^(r)a_(2)^(k), "log"_(e)a_(2)^(r)a_(3)^(k),"log"_(e)a_(3)^(r)a_(4)^(k)),("log"_(e)a_(4)^(r)a_(5)^(k), "log"_(e)a_(5)^(r)a_(6)^(k),"log"_(e)a_(6)^(r)a_(7)^(k)),("log"_(e)a_(7)^(r)a_(8)^(k), "log"_(e)a_(8)^(r)a_(9)^(k),"log"_(e)a_(9)^(r)a_(10)^(k)):}] = 0 ltbgt Then the number of elements is S, is

If a_(n)>1 for all n in N then log_(a_(2))a_(1)+log_(a_(3))a_(2)+....log_(a_(1))a_(n) has the minimum value of

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

If a_(1),a_(2),a_(3),.... are in G.P. with common ratio as r^(2) then log a_(1),log a_(2),log a_(3),.... will be in

Let the terms a_(1),a_(2),a_(3),…a_(n) be in G.P. with common ratio r. Let S_(k) denote the sum of first k terms of this G.P.. Prove that S_(m-1)xxS_(m)=(r+1)/r SigmaSigma_(i le itj le n)a_(i)a_(j)