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Let bᵢ > 1 for i = 1, 2,..., 101. Assume that logₑb₁, logₑb₂,..., logₑb₁₀₁ form an arithmetic sequence with a common difference of log₂. Also, let a₁, a₂,..., a₁₀₁ form an arithmetic sequence where a₁ = b₁ and a₅₁ = b₅₁. If t represents the sum b₁ + b₂ +... + b₅₁ and s represents the sum a₁ + a₂ +... + a₅₁, then which of the following is true?

1. s > t and a₁₀₁ > b₁₀₁
2. s > t and a₁₀₁ < b₁₀₁
3. s < t and a₁₀₁ > b₁₀₁
4. s < t and a₁₀₁ < b₁₀₁

Correct answer: s > t and a₁₀₁ < b₁₀₁

Solution

The sequences of logₑbᵢ and aᵢ are arithmetic, but their growth rates differ. The sum s of aᵢ exceeds the sum t of bᵢ because aᵢ grows faster. However, the last term a₁₀₁ is smaller than b₁₀₁ due to the logarithmic nature of bᵢ.

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