MATHCOUNTS 2026 State Sprint Round (Problem 24)In a certain sequence, a0=20a_0=20a0=20 andan=(an−1)2+1a_n=\sqrt{(a_{n-1})^2+1}an=(an−1)2+1for any positive integer nnn. What is the value of a500a_{500}a500?Related TopicsToolkit 5 — Telescoping seriesHints (6)Hint 1an2=an−12+1a_n^2=a_{n-1}^2+1an2=an−12+1Hint 2an2−an−12=1a_n^2-a_{n-1}^2=1an2−an−12=1Hint 3a5002−a4992=1a_{500}^2-a_{499}^2=1a5002−a4992=1a4992−a4982=1a_{499}^2-a_{498}^2=1a4992−a4982=1a4982−a4972=1a_{498}^2-a_{497}^2=1a4982−a4972=1⋮\vdots⋮a22−a12=1a_2^2-a_1^2=1a22−a12=1a12−a02=1a_1^2-a_0^2=1a12−a02=1Hint 4By adding all the equations,a5002−a02=500a_{500}^2-a_0^2=500a5002−a02=500Hint 5a5002=500+a02=500+202=900a_{500}^2=500+a_0^2=500+20^2=900a5002=500+a02=500+202=900Therefore,a500=±30a_{500}=\pm30a500=±30Hint 6a500=a4992+1≥0a_{500}=\sqrt{a_{499}^2+1}\ge0a500=a4992+1≥0Therefore,a500=30a_{500}=30a500=30Final Answer30