# arrow_back Prove by induction that for every natural number $n$, $n^{3}+2 n$ is divisible by 3

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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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Find the largest integer value of $n$ such that $1\times 3 \times 5 \times 7\times · · ·\times 31 \times 33 \times 35$ is divisible by $3^n$
Find the largest integer value of $n$ such that $1\times 3 \times 5 \times 7\times · · ·\times 31 \times 33 \times 35$ is divisible by $3^n$Find the largest integer value of $n$ such that $1\times 3 \times 5 \times 7\times · · ·\times 31 \times 33 \times 35$ is divisible by $3^n$ ...
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If $n$ is a natural number, the number $3^{4 n+4}-4^{3 n+3}$ is divisible by:
If $n$ is a natural number, the number $3^{4 n+4}-4^{3 n+3}$ is divisible by:If $n$ is a natural number, the number $3^{4 n+4}-4^{3 n+3}$ is divisible by: ...
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If $x^3 ‐ x = n$ and $x$ is a positive integer greater than $1$, is $n$ divisible by $8$?
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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.)
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.)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. ...
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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*}
Prove by induction that \begin{equation*} \sum_{i=1}^{n} i^3 = \left(\sum_{i=1}^{n} i\right)^2. \end{equation*}Prove by induction that $$\sum_{i=1}^{n} i^3 = \left(\sum_{i=1}^{n} i\right)^2.$$ ...