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

Question

Philoid · Accepted Answer

The A.M between a and b is (a + b)/2.

(an + bn)/(an-1 + bn-1) = (a + b)/2

2(an + bn) = (a + b)(an-1 + bn-1)

2an + 2bn = an + bn + ban-1 + abn-1

an + bn - ban-1 - abn-1 = 0

an - ban-1 + bn - abn-1 = 0

an-1(a – b) - bn-1(b – a) = 0

(an-1 - bn-1)(b – a) = 0

(an-1 - bn-1) = 0 or (b – a) = 0

an-1 = bn-1 or b = a

If a = b, then n can take any value.

If a = b, then an-1 = bn-1 = an-1/bn-1 = 1 = (a/b)n-1 = 1

As a = b, a/b = 1.

Therefore, (a/b)n-1 = 1 = n – 1 = 0 = n = 1

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