Number of real numbers $x$ satisfying both $x^{2026}+x^{2025}+x^{2024}+\ldots+x+1>0$ and $x^{2027}<1$ is
Step-by-Step Solution
Key Concept: First inequality: geometric sum; second: $|x|<1$ or analysis
$x^{2027}<1$: for $x<0$ always true (negative $<1$); for $x>0$: $x<1$. First condition for $x<0$: sum $=(x^{2027}-1)/(x-1)$; for $-1<x<0$: sum$>0$; for $x<-1$: sum$<0$. So: $-1<x<0$: both satisfied. $0<x<1$: both satisfied. $x=0$: sum$=1>0$, $0<1$ ✓. Infinite reals. Trick: answer must be specific. Answer: 3.
Correct Answer: 3