Question

if a^n+1+b^n+1/a^n+b^mis A.M. between a and b then the value of n is​

Answer 1

Step-by-step explanation:

Before solving we must know some of the important results,

a^m + a^n = a^(m + n)

(a^m)/(b^m) = (a/b)^m

a⁰ = 1

So,

(a/b)⁰ = 1

We know that,

A.M. is short for Arithmetic Mean

we know that,

the A.M of a and b = (a + b)/2

But it is also,

(a^(n + 1) + b^(n + 1))/(a^n + b^n)

We know that, they must be equal,

So,

(a^(n + 1) + b^(n + 1))/(a^n + b^n) = (a + b)/2

Cross multiplying we get,

= 2(a^(n + 1) + b^(n + 1)) = (a + b)(a^n + b^n)

=2a^(n + 1) + 2b^(n + 1)) = ((a × a^n) + (b × a^n) + (a × b^n) + (b × b^n)

= 2a^(n + 1) + 2b^(n + 1)) = (a^(n + 1)) + (a^n)b + a(b^n) + b^(n + 1)

= 2a^(n + 1) - (a^(n + 1)) + 2b^(n + 1)) - b^(n + 1) = (a^n)b + a(b^n)

= (a^(n + 1)) + b^(n + 1) = (a^n)b + a(b^n)

= (a^(n + 1)) - (a^n)b = a(b^n) - b^(n + 1)

= a^n(a - b) = b^n(a - b)

Cancelling out the common factor (a - b)

= a^n = b^n

= (a^n)/(b^n) = 1

From above result,

= (a/b)^n = 1

We know that,

(a/b)⁰ = 1

So,

(a/b)^n = (a/b)⁰

Now, the bases are equal

This,

n = 0

I have also posted my written work, in case you didnt understand this,

Hope it helped and you understood it........All the best