Geometry
# Topology

Suppose \(\{a_n\}\) is a sequence of real numbers such that \[a_{n+1} = \frac{1}{2}a_n + 3\] for each natural number \(n \geq 1.\) Which of these statements are true?

A. If \(\displaystyle \lim_{n \to \infty}a_n = L,\) then \(L = \frac{1}{2}L + 3.\)

B. \(\displaystyle \lim_{n \to \infty}a_n = 6.\)

Suppose \(\{a_n\}\) is a sequence of real numbers such that \(a_1 \neq -3\) and \[a_{n+1} = 2a_n + 3\] for each natural number \(n \geq 1.\) Which of these statements are true?

A. If \(\displaystyle \lim_{n \to \infty}a_n = L,\) then \(L = 2L + 3.\)

B. \(\displaystyle \lim_{n \to \infty}a_n = -3.\)

Suppose \(\{a_n\}\) is a sequence of real numbers such that \[a_{n+1} = \frac{1}{2}a_n + 3\] for each natural number \(n \geq 1.\) Which of these statements are true?

A. \(\{a_n\}\) is bounded.

B. \(\{a_n\}\) is monotone.

C. \(\{a_n\}\) converges.

**Definitions.**
A sequence \(\{x_n\}\) of real number is:

*bounded* if there is a real number \(M\) such that \(-M < x_n < M\) for all \(n.\)

*monotone* if \(x_n \leq x_{n+1}\) for all \(n,\) or \(x_n \geq x_{n+1}\) for all \(n.\)

*convergent* if there is a number \(L\) such that for every \(\epsilon > 0,\) there is a natural number \(N\) satisfying

\[n > N \Rightarrow |x_n - L| < \epsilon.\]

Suppose \(\{a_n\}\) is a sequence of real numbers such that \(a_1 \neq -3\) and \[a_{n+1} = 2a_n + 3\] for each natural number \(n \geq 1.\) Which of these statements are true?

A. \(\{a_n\}\) is bounded.

B. \(\{a_n\}\) is monotone.

C. \(\{a_n\}\) converges.

**Definitions.**
A sequence \(\{x_n\}\) of real number is:

*bounded* if there is a real number \(M\) such that \(-M < x_n < M\) for all \(n.\)

*monotone* if \(x_n \leq x_{n+1}\) for all \(n,\) or \(x_n \geq x_{n+1}\) for all \(n.\)

*convergent* if there is a number \(L\) such that for every \(\epsilon > 0,\) there is a natural number \(N\) satisfying

\[n > N \Rightarrow |x_n - L| < \epsilon.\]

Which of these statements are true?

A. Every **bounded, monotone** sequence of real numbers converges

B. Every **bounded** sequence of real numbers converges

C. Every **monotone** sequence of real numbers converges

**Definitions.**
A sequence \(\{x_n\}\) of real number is:

*bounded* if there is a real number \(M\) such that \(-M < x_n < M\) for all \(n.\)

*monotone* if \(x_n \leq x_{n+1}\) for all \(n,\) or \(x_n \geq x_{n+1}\) for all \(n.\)

*convergent* if there is a number \(L\) such that for every \(\epsilon > 0,\) there is a natural number \(N\) satisfying

\[n > N \Rightarrow |x_n - L| < \epsilon.\]

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