Discrete Structures, Prove by induction: $1+x+x^2+x^3+\dotsb+x^n=\frac{1-x^{n+1}}{1-x}$ assuming $x\neq1$
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Prove $ x^n-1=(x-1)(x^{n-1}+x^{n-2}+...+x+1)$
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Prove by induction: $\sin{x}+\sin{3x}+\dots+\sin{(2n-1)x}=\frac{\sin^2{nx}}{\sin{x}}$
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I'm having trouble with a proof by induction
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Prove by induction that $\forall n \in\mathbb N$, $3\mid4^{n}-1$.
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Using induction to prove $2^{n-1}(1 + a_1a_2\ldots a_n) \geq (1+a_1)(1+a_2)\ldots(1+a_n)$ for $a_i \geq 1$
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Prove by induction Fibonacci equality
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Prove that $x^n+x^{-n} \in \mathbf{N}$ if $x+\frac1x \in \mathbf{N}$
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How do I prove that $3$ divides $14^{2n}-1$ by induction?
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Proof by induction that $2 + 4 + 6 + \cdots + 2n = n(n+1)$
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Proving $1^3+ 2^3 + \cdots + n^3 = \left(\frac{n(n+1)}{2}\right)^2$ using induction
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Proof $\sum_{k=1}^n (-1)^{k+1} \binom{n}{k}\frac{1}{k} = H_n$ by induction
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Induction: prove using congruences
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Prove $x^n-1$ is divisible by $x-1$ by induction
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Proof that $x^n+\frac{1}{x^n}\in \mathbb Z$ by complete induction
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Summation inductional proof: $\frac{1}{1^2}+\frac{1}{2^2}+\frac{1}{3^2}+\ldots+\frac{1}{n^2}<2$
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Proving by induction that $1^3 + 2^3 + 3^3 + \ldots + n^3 = \left[\frac{n(n+1)}{2}\right]^2$
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Proof of inequality $2(\sqrt{n+1}-\sqrt{n}) < \frac{1}{\sqrt{n}} < 2(\sqrt{n} - \sqrt{n-1})$ using induction