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Prove using mathematical induction: The sequence a n is equals to the squareroot of 2a n-1 , a 1 equals to the squareroot of 2 is increasing; that is, a n ＜ a n+1

Prove using mathematical induction: 
The sequence an is equals to the squareroot of 2an-1 , a1 equals to the squareroot of 2 is increasing; that is, an ＜ an+1

Grade:12

1 Answers

Aditya Gupta
2081 Points
4 years ago
note that a2= sqrt(2*a1)= sqrt(2*2^0.5)= sqrt(2^3/2)= 2^(3/4), which is obviously greater than a1= 2^(1/2) as ¾ is greater than ½.
now, for some k assume ak less than a(k+1) is true. given a(k+1)= sqrt(2ak)
so our assumption becomes ak less than sqrt(2ak).....(1)
now, we wish to show that a(k+1) less than a(k+2)
but a(k+2)= sqrt(2a(k+1))= sqrt(2sqrt(2ak)).
so, we wish to show:
sqrt(2ak) less than sqrt(2sqrt(2ak)).
or equivalently
2ak less than 2sqrt(2ak) (simple squaring both sides)
or ak less than sqrt(2ak), which is true from (1).
hence proved
kindly APPROVE :=)

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