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inequality with sequence by arithmetic and geometric means

Source: Austrian Regional Competition For Advanced Students 2013, p3

February 19, 2020
inequalitiesArithmetic Mean-Geometric MeanSequence

Problem Statement

For non-negative real numbers a,a, bb let A(a,b)A(a, b) be their arithmetic mean and G(a,b)G(a, b) their geometric mean. We consider the sequence ⟨an⟩\langle a_n \rangle with a0=0,a_0 = 0, a1=1a_1 = 1 and an+1=A(A(an−1,an),G(an−1,an))a_{n+1} = A(A(a_{n-1}, a_n), G(a_{n-1}, a_n)) for n>0.n > 0. (a) Show that each an=bn2a_n = b^2_n is the square of a rational number (with bn≥0b_n \geq 0). (b) Show that the inequality ∣bn−23∣<12n\left|b_n - \frac{2}{3}\right| < \frac{1}{2^n} holds for all n>0.n > 0.
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