At sea level, P = 101 kPa, T = (25+273) = 298 K
At half the height of Kilimanjaro, you have the height = 5900/2 = 2950 m = 2.95 km. Find the temperature at this altitude.
Then, with the other information given, I don't think that the gas constant will be of any use, unless there is something more.
Just use the ideal gas equation.
PV = nRT
Initial conditions, P1, V1, T1, n1, R. Assuming density of air is constant, n1 and V1 is constant.
Giving: P1/T1 = k (where k is the constant nR/V)
And at half the mountain up, P2/T2 = k
Hence, we get P1/T1 = P2/T2
or
P1T2 = P2T1
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