Discrete Math Question: Induction. Define a sequence recursively as follows. x_1=1 and for n in N, x_{n+1}=sqrt{(x_n)^2+1//(x_n)^2}

Mariah Sparks 2022-07-17 Answered
Discrete Math Question: Induction
Define a sequence recursively as follows. x 1 = 1 and for n N , x n + 1 = ( x n ) 2 + 1 / ( x n ) 2
Prove using mathematical induction that for all n N, 1 x n n
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Answers (1)

tun1t2j
Answered 2022-07-18 Author has 13 answers
Step 1
Base Case, clear.
Induction Hypothesis: Assume that 1 x n n for all n m.
Step 2
Induction step: Consider x m . We note that
x m = ( x m 1 ) 2 + 1 ( x m 1 ) 2 ( x m 1 ) 2 + 1 ( m 1 ) 2 + 1 = m
The " 1 " portion is easier and is left to you to figure out. Notice that we did not need to find a closed form formula.

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