Find the value of n so that `(a^(n+1)+b^(n+1))/(a^(n)+b^n)` may be the geometric mean between a and b.

Explanation

`therefore (a^(n+1)+b^(n+1))/(a^(n)+b^(n))=(a+b)/(2)`
`implies (b^(n+1)[((a)/(b))^(n+1)+1])/(b^(n)[((a)/(b))+1])=(b)/(2)[((a)/(b))+1]`
`implies 2[((a)/(b))^(n+1)+1]=[((a)/(b))^(n)+1]((a)/(b)+1)`
Let `" " (a)/(b)=lambda`
` therefore 2lambda^(n+1)+2=(lambda^(n)+1)(lambda+1)`
` implies 2lambda^(n+1)+2=lambda^(n+1)+lambda^(n)+lambda +1`
` implies lambda^(n+1)-lambda^(n)-lambda + 1=0 implies(lambda ^(n)-1)(lambda-1)=0`
` lambda-1 ne 0 " " " " [therefore a ne b]`
`therefore lambda^(n)-1 =0 implies lambda^(n)=1=lambda^(0)`
`implies " " n=0`