Question

If is the A.M. between a and b , then find the value of n .

Answer 1

The arithmetic mean is a n + b n a n−1 + b n−1

The formula for the arithmetic mean between two terms, a and b is given by,

A.M.= a+b 2

Substitute the value of arithmetic mean as a n + b n a n−1 + b n−1 in the above expression.

a n + b n a n−1 + b n−1 = a+b 2 2( a n + b n )=( a+b )( a n−1 + b n−1 ) 2 a n +2 b n =a⋅ a n−1 +b⋅ a n−1 +a⋅ b n−1 +b⋅ b n−1 2 a n +2 b n = a n +b⋅ a n−1 +a⋅ b n−1 + b n

Further simplify the above expression.

a n + b n =b⋅ a n−1 +a⋅ b n−1 a n −b⋅ a n−1 =a⋅ b n−1 − b n a n−1 ( a−b )= b n−1 ( a−b ) a n−1 = b n−1

Further simplify the above expression.

( a b ) n−1 =1 ( a b ) n−1 = ( a b ) 0

Compare the powers of corresponding terms on both sides in the above expression.

n−1=0 n=1

Thus, the value of n is 1.