MathDB

2

Part of 2019 Centers of Excellency of Suceava

Problems(3)

Floor of an expression involving n-th square root

Source:

7/7/2020
For a natural number n≥2, n\ge 2, calculate the integer part of 1+nn−2/n. \sqrt[n]{1+n}-\sqrt {2/n} .
Dan Nedeianu
floor functionInteger Partnumber theoryalgebra
Known prime generating sequence has coprime terms

Source:

7/7/2020
Let (sn)n≥1 \left( s_n \right)_{n\ge 1 } be a sequence with s1 s_1 and defined recursively as sn+1=sn2−sn+1. s_{n+1}=s_n^2-s_n+1. Prove that any two terms of this sequence are coprime.
Dan Nedeianu
primenumber theoryGCD
Limit of sequence defined as a second order inequality recursion

Source:

7/7/2020
Let be two real numbers b>a>0, b>a>0, and a sequence (xn)n≥1 \left( x_n \right)_{n\ge 1} with x2>x1>0 x_2>x_1>0 and such that axn+2+bxn≥(a+b)xn+1, ax_{n+2}+bx_n\ge (a+b)x_{n+1} , for any natural numbers n. n. Prove that lim⁡n→∞xn=∞. \lim_{n\to\infty } x_n=\infty .
Dan Popescu
SequenceslimitConvergencereal analyssireal analysis