Question

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

Answer 1

We know the arithmetic mean of ‘a’ and ‘b’ is (a+b)/2
Hence (a+b)/2 = (aⁿ⁺¹+bⁿ⁺¹)/(aⁿ+bⁿ)
(a+b)/2 = (aⁿ x a + bⁿ x b)/(aⁿ+bⁿ)  
(aⁿa + aⁿb + bⁿa + bⁿb) = (2aⁿa+2bⁿb)
(aⁿb + bⁿa) = (aⁿa + bⁿb)
aⁿ(b – a)=bⁿ(b – a)
aⁿ = bⁿ
This is possible only when n = 0

Comments

Won't it be possible if n=2 in the last step?

---

No, as 'a' and 'b' are variables and won't be equal if n=2
If we take n=0 then a^0 becomes 1 and b^0 becomes 0 and hence we get LHS=RHS

---

Thank you sir.