Recent questions tagged induction

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Calculate, by using an appropriate formula: \(3-1+\frac{1}{3}-\ldots \ldots \ldots \ldots \ldots+\frac{1}{243}\)
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Given a finite sequence \(x_{1,1}, x_{2,1}, \ldots, x_{n, 1}\) of integers ( \(n \geq 2\) ), not all equal, define the sequences \(x_{1, k}, \ldots, x_{n, k}\) by\[x_{i, ...
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The Fibonacci sequence is defined by \(f_1=0, f_2=1\), and \(f_{n+2}=f_{n+1}+f_n\) for \(n \geq 1\). Prove that there is a strictly increasing arithmetic progression whos...
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Define a sequence \(a_{1}, a_{2}, \ldots\) by \(a_{1}=\frac{5}{2}\) and \(a_{n+1}=a_{n}^{2}-2\) for \(n \geq 1\). Give an explicit formula for \(a_{n}\) and prove it.
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We can also use induction in a definition. For example, the Fibonacci numbers is a sequence of numbers \(F_{0}, F_{1}, F_{2}, \ldots\) defined by \(F_{0}=F_{1}=1\) and\[F...
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What is the induction principle in mathematics?
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Let \(a_{0}=6\) and\[a_{n}=\frac{a_{n-1}}{1+a_{n-1}}\]for all \(n \geq 1\). Find \(a_{100}\).
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Let \(E\) be an open set in \(R^{n}\). The classes \(\mathcal{C}^{\prime}(E)\) and \(\mathcal{C}^{\prime \prime}(E)\) are defined in the text. By induction \(\mathcal{C}^...
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Prove by Induction :Prove \(3^{n}>2^{n}\) for all natural numbers \(n\).
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Let \(S_{N}:=1+\frac{1}{2}+\cdots+\frac{1}{N}\). Find an estimate for \(N\) so that \(S_{N}>100\).
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Prove by induction that for every natural number $n$, $n^{3}+2 n$ is divisible by 3
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An electroscope charged WITHOUT contacting a charged body is charged by:w) inductionx) conductiony) convectionz) insulation
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Show that $s_{n} \leq 2$ for all $n .$ (Hint: Use induction)
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Let $s_{1}=\sqrt{2}$ and $s_{n+1}=\sqrt{2+s_{n}}$ for $n=1,2,3, \cdots$.(i) Show that $\left\{s_{n}\right\}$ is an monotonically increasing sequence. (Hint: Use induction...
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What does electromagnetic induction depend on?
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What is electromagnetic induction in physics?
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Solve for $x$ in the equation $\left(2\sqrt{x}\right)^{n}\left(\dfrac{-1}{\sqrt{x}}\right) = -16$
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Define a sequence of real numbers by the rules\begin{equation*}s_0 = 0 \quad \text{and} \quad s_{n + 1} = \sqrt{3 + s_n} \enspace \text{for } n \geq 0\end{equation*}Show ...
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Consider the Fibonnaci sequence, given by F(0) = F(1) = 1, and for $n \geq 0$, F(n + 2) = F(n) + F(n + 1). Let $r = (\sqrt{5} + 1)/2$. Prove by induction that\begin{equat...
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Prove by induction that\begin{equation*}\sum_{i=1}^{n} i^3 = \left(\sum_{i=1}^{n} i\right)^2.\end{equation*}
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What is False Proof of Induction?
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What is the Generalized Principle of Induction?
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State Faraday’s Law of electromagnetic induction
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A material having uniform magnetization $M$ is in the form of a thin disk with radius $a$ and thickness $h \leq a$. $\overrightarrow{\rm M}$ points along the axis of th...
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What is electromagnetic induction?
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How can companies induct new employees as fast as possible to reduce costs?
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