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1

Part of 2005 Czech And Slovak Olympiad III A

Problems(1)

for some k > 1: x_{k-1}y_{k-1} = 42, x_ky_k = 30, and x_{k+1}y_{k+1} = 16

Source: Czech and Slovak MO, III A, 2005 p1

1/12/2020
Consider all arithmetical sequences of real numbers (xi)∞=1(x_i)^{\infty}=1 and (yi)∞=1(y_i)^{\infty} =1 with the common first term, such that for some k>1,xk−1yk−1=42,xkyk=30k > 1, x_{k-1}y_{k-1} = 42, x_ky_k = 30, and xk+1yk+1=16x_{k+1}y_{k+1} = 16. Find all such pairs of sequences with the maximum possible kk.
arithmetic sequenceSequencealgebramaximumrecurrence relation