Positive numbers a_n satisfy 8/a_(n+1)=7/a_n+a_n²/343,n≥1,a₁=3,and a_n<7 for n≥2. I:a_n is increasing. II:a_n converges to a value in [3,7]. Which hold?
Rewrite a_(n+1)/a_n=8/[7+(a_n/7)³]. Since 0<a_n<7,the denominator is below 8,so the ratio exceeds 1. The sequence is increasing and bounded above by 7,hence converges in [3,7]. In fact its positive limit solves 8=7+(L/7)³,so L=7. **C**.
More Sequences and Series questions
GATE 2022 ST Q37– Which series converge conditionally? S₁=Σ (n=0)∞(−1)n n/(n²+4); S₂=Σ…