InterviewSolution
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InterviewSolution
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Using binomial theorem, prove that 23n – 7n – 1 is divisible by 49, where n ∈ N. |
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Answer» Given as 23n – 7n – 1 Therefore, 23n – 7n – 1 = 8n – 7n – 1 Now, 8n – 7n – 1 8n = 7n + 1 = (1 + 7) n = nC0 + nC1 (7)1 + nC2 (7)2 + nC3 (7)3 + nC4 (7)2 + nC5 (7)1 + … + nCn (7) n 8n = 1 + 7n + 49 [nC2 + nC3 (71) + nC4 (72) + … + nCn (7) n-2] 8n – 1 – 7n = 49 (integer) Therefore now, 8n – 1 – 7n is divisible by 49 Or 23n – 1 – 7n is divisible by 49. Thus proved. |
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