Expansion of (1+\sqrt2)^n I was asked to show that

Helen Lewis

Helen Lewis

Answered question


Expansion of (1+2)n
I was asked to show that nN there exist a pN such that
I used induction but it wasn't fruitful, so I tried to use the binomial expansion of (1+2)n but it seems I lack some insight to go further.
Any hint is welcomed.

Answer & Explanation



Beginner2022-01-06Added 35 answers

The binomial formula shows you that
for some integers an,bn.
But, the same binomial formula shows you that (convince yourself of this)
for the same integers an,bn.
Then comes the hint: Calculate both
and compare.
You will get that a2n2b2n=(1)n, so a2n and 2b2n differ from each other by one, and p will be the larger of the two.
Mary Goodson

Mary Goodson

Beginner2022-01-07Added 37 answers

This is a bit more roundabout than Jyrkis


Expert2022-01-11Added 669 answers

It's reduced to show that the following expression––––––––––––––––––––– is an integer:

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