Let a(1), a(2), a(3)……, a(10) " be in GP...

Let `a_(1), a_(2), a_(3)……, a_(10) " be in GP with " a_(i) gt1 " 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` Then the number of elements is S, is

infinitely many

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