Compressors, Gas Turbines and Jet Engines
The assumption made in two stage compression with inter-cooler is that

There is no pressure drop in the inter-cooler
The suction and delivery of air takes place at constant pressure
All of the listed here
The compression in both the cylinders is polytropic

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Compressors, Gas Turbines and Jet Engines
In n₁ and n₂ are the indices of compression for the first and second stage of compression, then the ratio of work-done on the first and second stages (W₁/W₂) with perfect intercooling is given by

W₁/W₂ = n₁(n₂ - 1)/n₂(n₁ - 1)
W₁/W₂ = n₂/n₁
W₁/W₂ = n₂(n₁ - 1)/n₁(n₂ - 1)
W₁/W₂ = n₁/n₂

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