Question

Let b_i >1 for i = 1, 2, ..., 101. Suppose log_e b₁, log_e b₂, , log_e b₁₀₁ are in Arithmetic Progression (A. P.) with the common difference log_e 2. Suppose a₁, a₂, a₁₀₁ are in A.P. such that a₁ b₁ and as₁ b₅₁. If = t = b₁ + b₂ +...+ b₅₁ and s = a₁ + a₂ + ... + a₅₁· then 

(A) s > t and a₁₀₁ > b₁₀₁ 

(B) s > t and a₁₀₁ < b₁₀₁ 

(C) s < t and a₁₀₁ > b₁₀₁ 

(D) s < t and a101 < b₁₀₁

Answer 1

(B) s > t and a₁₀₁ < b₁₀₁ 

a₂, a₃, ....., a50 are Arithmetic Means and b₂, b₃, ....., b50 are Geometric Means between a₁(= b₁) and a₅₁(= b₅₁) 

Hence b₂ < a₂, b₃ < a₃ ..... 

⇒ t < S 

Also a₁, a₅₁, a101 is an Arithmetic Progression and b₁, b₅₁, b₁₀₁ is a Geometric Progression 

Since a₁ = b₁ and a₅₁ = b₅₁ 

⇒ b₁₀₁ > a₁₀₁