Question

If $ \dfrac{{{a^n} + {b^n}}}{{{a^{n - 1}} - {b^{n - 1}}}} $ is the A.M. between a and b, then find the value of n.

Answer 1

Hint: In order to solve this question we will first write the given A.M in question equal to the general known formula for calculation then we will make the proper calculations then after the appropriate calculation we will make the factors the we will find the value of n from that factor.

Complete step by step solution:
For solving this question we will first put the given A.M equal to the known general formula of the known for the A.M:
According to the equation:
 $ \dfrac{{{a^n} + {b^n}}}{{{a^{n - 1}} - {b^{n - 1}}}} = \dfrac{{a + b}}{2} $
Now on cross multiplication:
 $ 2\left( {{a^n} + {b^n}} \right) = \left( {a + b} \right)\left( {{a^{n - 1}} + {b^{n - 1}}} \right) $
Now on further simplification we will get:
 $ 2{a^n} + 2{b^n} = a\left( {{a^{n - 1}} + {b^{n - 1}}} \right) + b\left( {{a^{n - 1}} + {b^{n - 1}}} \right) $
We will simplify it at the final stage:
 $ 2{a^n} + 2{b^n} = {a^n} + a{b^{n - 1}} + {a^{n - 1}}b + {b^n} $
By taking all the values to L.H.S we will get:
 $ {a^n} - {a^{n - 1}}b + {b^n} - a{b^{n - 1}} = 0 $
Now taking the common $ {a^{n - 1}} $ from first two terms and $ {b^{n - 1}} $ from last two terms we will get:
 $ {a^{n - 1}}\left( {a - b} \right) + {b^{n - 1}}\left( {a - b} \right) = 0 $
Now taking a-b common from both the terms we will get:
 $ \left( {a - b} \right)\left( {{a^{n - 1}} - {b^{n - 1}}} \right) = 0 $
Now for this equation to satisfy either one of two should be correct:
 $ a - b = 0 $ or $ \left( {{a^{n - 1}} - {b^{n - 1}}} \right) = 0 $
So from first one we will get:
 $ a = b $
Which is not possible for any A.P
Now second one;
 $ {a^{n - 1}} = {b^{n - 1}} $
On substituting it we will get:
 $ \dfrac{{{a^{n - 1}}}}{{{b^{n - 1}}}} = 1 $
Now writing it into new form we already know that the power of any number to be zero is equal to 1.
 $ {\left( {\dfrac{a}{b}} \right)^{n - 1}} = {\left( {\dfrac{a}{b}} \right)^0} $
When we will compare it we will get:
 $ n - 1 = 0 $
From here we will get the value of n = 1.
So, the correct answer is “n = 1”.

Note: While solving these types of problems we should keep in mind that we are making the appropriate changes for finding the factors, because if we are taking any other step either taking any other common we cannot the proper factor so this will be insolvable.