Exercise · ZW-MONOTONE-SEQUENCE-007 · ★★☆ Intermediate
If \((a_n)\) is increasing and \(a_1=3\), what always holds?
Choose one option.
Show answer
- A. \(a_n\ge3\) for all \(n\)
Explanation
Increasing means \(a_n\le a_{n+1}\), so \(a_1\le a_n\) for all \(n\). Strict inequality is not guaranteed (it may be constant), nor is an upper bound (e.g. \(a_n=n+2\)).
Review the concept: Monotone sequence
A real sequence \((a_n)\) is monotone when, as the index \(n\) grows, its terms either never decrease or never increase.
\[a_n\le a_{n+1}\ \ \forall n\quad\text{or}\quad a_n\ge a_{n+1}\ \ \forall n\]
More practice problems: monotone sequence
- ZW-MONOTONE-SEQUENCE-001 The sequence a_n=2n+5 is:
- ZW-MONOTONE-SEQUENCE-002 The sequence a_n=1n is:
- ZW-MONOTONE-SEQUENCE-003 The sequence a_n=(-1)^n is:
- ZW-MONOTONE-SEQUENCE-004 The sequence a_n=(n)/(n+1) is:
- ZW-MONOTONE-SEQUENCE-005 Let a_n=n^2-6n. Which statement is correct?
- ZW-MONOTONE-SEQUENCE-006 Let a_1=-4 and a_n+1=(a_n)/(2). The sequence is:
- ZW-MONOTONE-SEQUENCE-008 The sequence a_n=(2n)/(n^2+1) is:
- ZW-MONOTONE-SEQUENCE-009 If (a_n) and (b_n) are increasing, which sequence is always increasing?