Q.204.

Question 204.
Solve the following for n (n > 0):
(n^3 - 3)/n^3 + (n^3 - 4)/n^3 + (n^3 - 5)/n^3 + (n^3 - 6)/n^3 + ... + 5/n^3 + 4/n^3 + 3/n^3 = 169.

Answer 204.
(n^3 - 3)/n^3 + (n^3 - 4)/n^3 + (n^3 - 5)/n^3 + (n^3 - 6)/n^3 + ... + 5/n^3 + 4/n^3 + 3/n^3 = 169

=> (1/n^3) * [3 + 4 + 5 + .... + (n^3 - 3)] = 169
=> (1/n^3) * [ (1/2) (n^3 - 3) (n^3 - 2) - 3] = 169
=> 338 n^3 = (n^3 - 3) (n^3 - 2) - 6
=> 338 n^3 = n^6 - 5n^3
=> n^3 (n^3 - 343) = 0
=> n = 0 or n^3 = 343 => n = 7
n = 0 is not possible as n appears in the denominator of all terms
=> n = 7 is the answer.

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