Let bᵢ > 1 for i = 1, 2, ..., 101. Suppose logₑb₁, logₑb₂, ..., logₑb₁₀₁ are in Arithmetic Progression (A.P) with the common difference logₑ2. Suppose a₁, a₂, ... a₁₀₁ are in A.P. such that a₁ = b₁ and a₅₁ = b₅₁. If t = b₁ + b₂ + ... + b₅₁ and s = a₁ + a₂ + ... + a₅₁, then what is the relationship between s and t, and a₁₀₁ and b₁₀₁?

logeb1,logeb2,logeb3,..logeb101are in A.P.b1,b2,b3,...,b101are in G.P.Given :loge(b2)−loge(b1)=loge(2)⇒b2b1=2=r(common ratio of G.P. )a1,a2,a3,.a101are in A.P.a1=b1=ab1+b2+b3+b51=t,S=a1+a2+..+a51t=sum of51terms of G.P.=b1(r51𕒵)r𕒵=a(251𕒵)2𕒵=a(251𕒵)s=sum of51terms of A.P=512]2a1+(n𕒵)d]=512(2a+50d)Givena51=b51a+50d=a(2)5050d=a(250𕒵)Hence,⇒s=a(51.249+512)s=2(4.249+47.249+512)⇒s=a((251𕒵)+47.249+532)s−t=a(47.249+532)Clearlys>ta101=a1+100d=a+2a.2502a=a(251𕒵)b101=b1r100=a.2100Hence :b101>a101