maths help

Question

For what value n (a^n+1 + b^n+1)/(a^n+b^n) is in Gm?

Answer 1

Let’s explore the expression (\frac{{a^{n+1} + b^{n+1}}}{{an + b^n}}) and find out for which value of (n) it represents the arithmetic mean of (a) and (b).

First, let’s simplify the expression: [ \frac{{a^{n+1} + b^{n+1}}}{{an + b^n}} ]

We know that the arithmetic mean of two numbers (a) and (b) is given by: [ \text{AM} = \frac{{a + b}}{2} ]

For the given expression to be the arithmetic mean of (a) and (b), we need: [ \frac{{a^{n+1} + b^{n+1}}}{{an + b^n}} = \frac{{a + b}}{2} ]

Multiplying both sides by (2(a^n + b^n)): [ 2(a^{n+1} + b^{n+1}) = a(a^n + b^n) + b(a^n + b^n) ]

Expanding the right side: [ 2a^{n+1} + 2b^{n+1} = (a^{n+1} + ab^n) + (b^{n+1} + ba^n) ]

Rearranging terms: [ a^{n+1} - ab^n + b^{n+1} - ba^n = 0 ]

Factoring out (a^n) and (b^n): [ a^n(b - 1) + b^n(a - 1) = 0 ]

For this equation to hold, either (a^n = 0) (which is not possible since (a) is nonzero) or: [ b - 1 = 0 \quad \text{and} \quad a - 1 = 0 ]

Solving for (a) and (b): [ a = 1 \quad \text{and} \quad b = 1 ]

Therefore, the given expression represents the arithmetic mean of (a) and (b) when (n = 0). 🌟¹²

Feel free to ask if you have any more questions! 😊