Given n 4 n for a fixed positive integer n ≥ 2, prove that (n + 1) 4 n + 1.

Question

Navjyot Kalra · Answer

Sol. Given that n 4 < 10 n for n for a fixed + ve integer n &ge;2. To prove that (n + 1) 4 , 10 n + 1 Proof : Since n 4 < 10 n &rArr; 10n 4 < 10 n + 1 . . . . . . . . . . . . . . . . . . (1) So it is sufficient to prove that (n + 1) 4 < 10n 4 Now (n + 1/n) 4 = (1 + 1/n) 4 &le; (1 + 1/2) 4 [∵ n &ge; 2] = 81/16 < 10 &rArr; (n + 1) 4 < 10n 4 . . . . . . . . . . . . . . . . . . . . . . . .(2) From (1) and (2), (n + 1) 4 < 10 n + 1

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Given n 4 n for a fixed positive integer n ≥ 2, prove that (n + 1) 4 n + 1.

Given n⁴ ⁿ for a fixed positive integer n ≥ 2, prove that (n + 1)⁴ ^{n + 1.}

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Given n 4 n for a fixed positive integer n ≥ 2, prove that (n + 1) 4 n + 1.

Given n4 n for a fixed positive integer n ≥ 2, prove that (n + 1)4 n + 1.

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Given n⁴ ⁿ for a fixed positive integer n ≥ 2, prove that (n + 1)⁴ ^{n + 1.}