Answers
Lets simplify our problem and take n=2.
let a1 = 2 , then a2 = 1/2
or in general if a1 = k, then a2 = 1/k, such that a1.a2 = 1
Now if we take the function f = (1+a1+(a1^2))(1+a2+(a2^2))
Put a1 = k, a2 = 1/k , we get
f = ((1+k+(k^2))/k)^2
Now, we can extend this model to n.
All we have to do is, understand that a1*a2*a3*…*an = 1,
whenever numerator is equal to denominator.
for example, consider the sequence
2,3,4,1/2,1/3,1/4, this if taken the product leaves 1.
2*3*4 = 24 and 1/2*1/3*1/4 = 1/*24.
we can assume the numerator equal denominator to be K
thus, f = (((1+K+(K^2))/k)^n
here, we see its an increasing function and given the domain of positive real numbers this function has a minimum at k = 1. The minimum value of the function will be 3^n.
Minimum of f = 3^n.
I Hope it helps, I am learning to type in Latex. :)
Step-by-step explanation:
Refer to the attachmmet