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junior college 1 | H2 Maths
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Hi, im really not how to do this mathematical induction question. Pls help me, please.
Date Posted: 5 years ago
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Let p(n) be the statement ln n < n for some positive integer n. Trivially, p(1) is true since ln 1 = 0 < 1. We established a base case for the induction principle.
Suppose p(k) is true for some positive integer k (induction hypothesis). This means that ln k < k.
We note that k+1 <= 2k. Since f(x) = ln(x) is an increasing function i.e. as x increases, f(x) also increases, therefore ln( k + 1) <= ln(2k) < ln(k * e) = ln k + ln e. By the induction hypothesis, we know that ln k + ln e < k + ln e = k + 1. Thus, p(k+1) is true.
Since p(k) implies p(k+1) for some positive integer k, and p(1) is true, hence p(n) is true for all positive integers n.
Hope this helps
Suppose p(k) is true for some positive integer k (induction hypothesis). This means that ln k < k.
We note that k+1 <= 2k. Since f(x) = ln(x) is an increasing function i.e. as x increases, f(x) also increases, therefore ln( k + 1) <= ln(2k) < ln(k * e) = ln k + ln e. By the induction hypothesis, we know that ln k + ln e < k + ln e = k + 1. Thus, p(k+1) is true.
Since p(k) implies p(k+1) for some positive integer k, and p(1) is true, hence p(n) is true for all positive integers n.
Hope this helps
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