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Math Help - mathematical induction recursion

  1. #1
    Senior Member polymerase's Avatar
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    mathematical induction recursion

    How do I prove by induction,

    If t1 = 1, tn+1 = 1 + (1 + tn) prove a) tn < 3 for all n N
    b) tn+1 > tn for all n N
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  2. #2
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    Hello, polymerase!

    I'll help you with part (a) . . .


    If t_1 = 1,\;t_{n+1} = 1 + \sqrt{1 + t_n}, prove by induction:

    a) t_n < 3 for all n \in N

    Verify S(1). . Is t_1 < 3?
    . . t_1 = 1 < 3 . . . yes!


    Assume S(k) is true: . t_k \:< \:3

    Add 1 to both sides: .  1 + t_k\:<\:4

    Take the square root of both sides: . \sqrt{1 + t_k} \:<\:2

    Add 1 to both sides: . \underbrace{1 + \sqrt{1 + t_k}}_{\text{This is }t_{k+1}} \:<\:3

    Therefore: . t_{k+1}\:<\:3
    . . The inductive proof is complete.

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