If `(a^n+b^n)/(a^(n-1)+b^(n-1))` is the A.M. between a and b, then find the value of n.

Answer

N/a

Explanation

Arithmetic mean of a and `b=(a+b)/(2)`
`"According to the problem, "(a^(n)+b^(n))/(a^(n-1)+b^(n-1))=(a+b)/(2)`
`rArr" "2a^(n)+2b^(n)=a^(n)+a^(n-1)b+ab^(n-1)+b^(n)`
`rArr" "a^(n)-a^(n-1)b-ab^(n-1)+b^(n)=0`
`rArr" "a^(n-1)(a-b)-b^(n-1)(a-b)=0`
`rArr" "(a-b)(a^(n-1)-b^(n-1))=0`
`rArr" "a^(n-1)-b^(n-1)=0 or a-b=0`
`rArr" "a^(n-1)=b^(n-1) or a=b`
`rArr" "((a)/(b))^(n-1)=((a)/(b))^(0)`
`rArr" "n-1=0" "(becauseaneb)`
`rArr" "n=1`