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a_(1) = 1 \\ a_(n + 1) = \sqrt[]{1 + a_(n)}

Prove by mathematical induction that the sequence is increasing​

User Rdasxy
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2 Answers

10 votes
10 votes

Trivially, we have


a_2 = √(1+a_1) = \sqrt2 > 1 = a_1

so the base case is satisfied.

Assume
a_k > a_(k-1), so


a_k = \sqrt{1+a_(k-1)} > a_(k-1)

We want to use this to show
a_(k+1) > a_k.
\sqrt x is an increasing function, so from our assumption it follows that


a_(k+1) = √(1 + a_k) > \sqrt{1 + a_(k-1)} = a_k

QED

User Garrett Motzner
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3.1k points
9 votes
9 votes

Answer + Step-by-step explanation:


a_(1)=1


a_(2)=\sqrt{1+a_(1)} =√(1+1) = √(2)

then


a_(2) \geq a_(1)


\text{suppose}\ a_(n+1) \geq a_(n)\ \text{and prove that} \ a_(n+2)\geq a_(n+1)


a_(n+2)=\sqrt{1+a_(n+1)} \geq \sqrt{1+a_(n)} = a_(n+1)

Then


a_(n+2) \geq a_(n+1)

Then ,according to the principal of mathematical induction:

the sequence (an) is increasing​.

User Joe Fernandez
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3.7k points