Solve all parts (a,b,c,d,e) without copying others because their answers are wrong, otherwise i will downvote. The issue

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Solve all parts (a,b,c,d,e) without copying others because their answers are wrong, otherwise i will downvote.The issue 4.7 Consider a chain of N + 1 m-mass atoms that is linear in shape, with harmonic forces acting between nearest neighbours. These forces could be represented by N springs with a K-spring constant. Considering the atoms at either end of the chain to be stationary, determine the dispersion relation for longitudinal waves on the chain. Discover the density of states and an expression for the internal energy at temperature T for the situation where N is very large. In the high and low temperature ranges, evaluate the internal energy specifically.