Sequences Quiz

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1. Find a formula for the $n$th term of the sequence $\{25, -1/5, 1, -5, \ldots\}_{n=1}^\infty$.

   (a) $\dfrac{5}{(-5)^{n-1}}$
   (b) $\dfrac{25}{(-5)^{n-1}}$
   (c) $\dfrac{-5}{5^{n-1}}$
   (d) $\dfrac{-25}{5^{n-1}}$

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2. Find the limit of $a_n = 2^n \cdot 3^{-n}$.

   (a) $0$
   (b) $1$
   (c) $\infty$
   (d) $\dfrac{2}{\sqrt{3}}$

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3. Is $a_n = n + \sin{n}$ bounded? What is $\lim_{n \to \infty} a_n$?

   (a) Bounded between $-1$ and $\dfrac{\pi}{4}$
   (b) Bounded between $0$ and $1$
   (c) Unbounded, and $\lim_{n \to \infty} a_n = \infty$
   (d) Unbounded, and $\lim_{n \to \infty} a_n = -\infty$

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4. Does $a_n = \sqrt{\dfrac{n+1}{9n+1}}$ converge or diverge? If it converges, find the limit.

   (a) Converges to $\dfrac{1}{9}$
   (b) Converges to $\dfrac{1}{3}$
   (c) Converges to $\dfrac{1}{\sqrt{3}}$
   (d) Diverges

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5. Does $a_n = \cos{(n\pi)}(-1/2)^n$ converge? If it converges, find the limit.

   (a) $\dfrac{\pi}{2}$
   (b) $0$
   (c) $1$
   (d) $\infty$

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