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Given n4 n for a fixed positive integer n ≥ 2, prove that (n + 1)4 n + 1.
Sol. Given that n4 < 10n for n for a fixed + ve integer n ≥2. To prove that (n + 1)4 , 10n + 1 Proof : Since n4 < 10n ⇒ 10n4 < 10n + 1 . . . . . . . . . . . . . . . . . . (1) So it is sufficient to prove that (n + 1)4 < 10n4 Now (n + 1/n)4 = (1 + 1/n)4 ≤ (1 + 1/2)4 [∵ n ≥ 2] = 81/16 < 10 ⇒ (n + 1)4 < 10n4 . . . . . . . . . . . . . . . . . . . . . . . .(2) From (1) and (2), (n + 1)4 < 10n + 1
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