Solution to Let the sequence Tn be defined by T1 = T2 = T3 = 1 and … - Sikademy
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Let the sequence Tn be defined by T1 = T2 = T3 = 1 and Tn = Tn-1 + Tn-2 + Tn-3 for n ≥ 4. Use induction to prove that Tn < 2n for n ≥ 4

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Let P(n) be the proposition that T_n<2^n.

Basis Step

P(4) is true, because T_4=T_1+T_2+T_3=1+1+1=3<16=2^4.

Inductive Step

We assume that


T_{k-1}<2^{k-1}

T_{k-2}<2^{k-2}

T_{k-3}<2^{k-3}


T_k=T_{k-1}+T_{k-2}+T_{k-3}<2^k

Under this assumption


T_{k+1}=T_{k}+T_{k-1}+T_{k-2}<2^{k}+2^{k-1}+2^{k-2}

=2^{k+1}(\dfrac{1}{2}+\dfrac{1}{4}+\dfrac{1}{8})=2^{k+1}(\dfrac{7}{8})<2^{k+1}

P(k + 1) is true under the assumption that P(k) is true. This completes the inductive step.

We have completed the basis step and the inductive step, so by mathematical induction we

know that P(n) is true for all n\geq4. That is, we have proved that


T_n=T_{n-1}+T_{n-2}+T_{n-3}<2^n, n\geq 4


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