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Show the identity for sequence T_n - ISL 1971

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September 22, 2010
algebraSequencerecurrence relationLinear RecurrencesIMO Shortlist

Problem Statement

Let Tk=k−1T_k = k - 1 for k=1,2,3,4k = 1, 2, 3,4 and T2k−1=T2k−2+2k−2,T2k=T2k−5+2k(k≥3).T_{2k-1} = T_{2k-2} + 2^{k-2}, T_{2k} = T_{2k-5} + 2^k \qquad (k \geq 3). Show that for all kk, 1 + T_{2n-1} = \left[ \frac{12}{7}2^{n-1} \right]   \text{and}   1 + T_{2n} = \left[ \frac{17}{7}2^{n-1} \right], where [x][x] denotes the greatest integer not exceeding x.x.
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