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9 gru 2014 · How can I prove that $$1^3+ 2^3 + \cdots + n^3 = \left(\frac{n(n+1)}{2}\right)^2$$ for all $n \in \mathbb{N}$? I am looking for a proof using mathematical induction. Thanks
Let P(n) be “Either n < 5 or n2 < 2n.” P(0) is trivially true. P(1) is trivially true, so P(0) → P(1) P(2) is trivially true, so P(1) → P(2) P(3) is trivially true, so P(2) → P(3) P(4) is trivially true, so P(3) → P(4) We explicitly proved P(5), so P(4) → P(5) For any n ≥ 5, we explicitly proved that P(n) → P(n + 1).
16 kwi 2024 · Ex 4.1,2: Prove the following by using the principle of mathematical induction 13 + 23 + 33+ + n3 = ( ( +1)/2)^2 Let P (n) : 13 + 23 + 33 + 43 + ..+ n3 = ( ( +1)/2)^2 For n = 1, L.H.S = 13 = 1 R.H.S = (1(1 + 1)/2)^2= ((1 2)/2)^2= (1)2 = 1 Hence, L.H.S. = R.H.S P(n) is true f
In mathematics, the harmonic series is the infinite series formed by summing all positive unit fractions: The first terms of the series sum to approximately , where is the natural logarithm and is the Euler–Mascheroni constant.
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27 cze 2017 · using the method of proof by induction. this involves the following steps. ∙ prove true for some value, say n = 1. ∙ assume the result is true for n = k. ∙ prove true for n = k + 1. n = 1 → LH S = 12 = 1. and RHS = 1 6 (1 + 1)(2 +1) = 1. ⇒result is true for n = 1. assume result is true for n = k. assume 12 +22 +.... +k2 = 1 6 k(k +1)(2k +1)
