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Problem Solving Set 21 24 July 2012 1. (a) A sequence x 1 ,x 2 ,... of real numbers satises x n+1 = x n cos x n for all n 1. Does it follow that this sequence con- verges for all initial values x 1 ? (b) A sequence y 1 ,y 2 ,... of real numbers satises y n+1 = y n sin y n for all n 1. Does it follow that this sequence con- verges for all initial values y 1 ? 2. Show that in any collection of 52 distinct positive integers there are two distinct numbers whose sum or difference is divisible by 100.

Problems 21

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  • Problem SolvingSet 21

    24 July 2012

    1. (a) A sequence x1, x2, . . . of real numbers satises

    xn+1 = xn cosxn

    for all n 1. Does it follow that this sequence con-verges for all initial values x1?

    (b) A sequence y1, y2, . . . of real numbers satises

    yn+1 = yn sin yn

    for all n 1. Does it follow that this sequence con-verges for all initial values y1?

    2. Show that in any collection of 52 distinct positive integersthere are two distinct numbers whose sum or difference isdivisible by 100.