Question

Q1)Is there any identity for :
${a}^{n} + {b}^{n}$

Q2) State the identity


Q3)If there is any identity for this , then evaluate:
${1}^{1997} + {1996}^{1997}$
​

Answer 1

 

There is an identity for the above problem .

$a^n+b^n=(a+b)(a^{n-1}b+........ab^{n-1})$

$a^n+b^n$ is always divisible by a - b .

$a^n+b^n$ is divisible by a + b provided that a + b is odd .

Here 1 + 1996 is odd .

Hence that has to be divisible by 1997 .

Evaluation of identity is of no use .

It will be very complex and cannot be done using calculator howsoever you know the identity.

Still trying :

$1^{1997}=1$

$1996^{1997}$ is almost impossible to find.

But in exams you should know the identity that a + b is a factor of $a^n+b^n$ and you dont need to evaluate.

Answer 2

There is an identity for the above problem .

a^n+b^n=(a+b)(a^{n-1}b+........ab^{n-1})

a^n+b^n is always divisible by a - b .

a^n+b^n is divisible by a + b provided that a + b is odd .

Here 1 + 1996 is odd .

Hence that has to be divisible by 1997 .

Evaluation of identity is of no use .

It will be very complex and cannot be done using calculator howsoever you know the identity.

Still trying :

1^{1997}=1

1996^{1997} is almost impossible to find.

But in exams you should know the identity that a + b is a factor of a^n+b^nand you dont need to evaluate.