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13 gru 2015 · Prove by induction the following formula: 1 + 3 +32 + ⋯ +3n−1 = 1 2(3n − 1). Solution. Proof by induction is given in three following steps: Base: assume n = 2, then 1 + 3 = 4 = 1 2(32 − 1), so the formula is correct. Hypothesis: assume formula holds for some positive integer n.
Hence, the statement holds for the $n + 1$ case. Thus by the principle of mathematical induction $1^3 + 2^3 + 3^3 + \cdots + n^3 = (1 + 2 + 3 + \cdots + n)^2$ for each $n \in \mathbb N$.
In example to get formula for $1^2+2^2+3^2+...+n^2$ they express $f(n)$ as: $$f(n)=an^3+bn^2+cn+d$$ also known that $f(0)=0$, $f(1)=1$, $f(2)=5$ and $f(3)=14$ Then this values are inserted into fu...
5 mar 2014 · Mathematical Induction: Example 11) Prove that 1^3 + 2^3 +3^3 ... n^3 = n^2 {(n+1)^2} /4
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