Track / Stage 4 / 287 of 340 #547 of 2000

Problem 547

AMC 12 late, AIME early

Algebra Difficulty 5.0 Find the answer

10. If $\left\{a_{n}\right\}$ is an increasing sequence, and for any positive integer $n, a_{n}=n^{2}+\lambda n$ always holds, then the range of the real number $\lambda$ is $\qquad$ .

Your answer

Official solution

10. $(-3,+\infty)$ .
$a_{n+1}-a_{n}=2 n+1+\lambda>0$ , for all $n \in \mathbf{N}_{+}$ to hold, it is sufficient that $3+\lambda>0$ , then $\lambda>-3$ .

Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic, difficulty and ordering added by this site.