I've got stuck on an induction problem, any help would be great
The terms of the sequence are: √5, √(5√5), √(5(√5(√5))),....
Use induction to show that for natural number n, 0<A(n)<5
Show by induction that
So, being < 5 causes all subsequent terms of the sequence to be < 5 also.
A non-inductive proof is as follows...
The index is a geometric series, first term=1, common ratio = 0.5 (for the part in brackets)
so the index of 5 is 1.
Hence cannot reach 5 as a finite number of terms will not reach the sum to infinity as all the terms are positive.