By using PMI, Prove that, x2n- y2nis divisible by x+y - Maths - Principle of Mathematical Induction

Question

By using PMI,
Prove that, x2n- y2nis divisible by x+y

Pooja Khanna · Accepted Answer

Let the given statement be Pn
 Then,Pn : x2n-y2n is divisible by x+y
When n=1, the given statement becomes:x2-y2 is divisible by x+y which is clearly true
So, P1 is true Let Pk be true
 Then,Pk : x2k-y2k is divisible by x+y        
1Now, x2k+1-y2k+1=x2k x2-y2k y2=x2k x2-x2y2k+x2y2k-y2k
y2=x2x2k-y2k+y2kx2-y2=x2x2k-y2k+y2kx+yx-y, which is divisible by x+y      using 1⇒Pk+1 : x2k+1-y2k+1 is divisible by x+y 
⇒Pk+1 is true, whenever Pk is true
Thus, P1 istrue and Pk+1 is true, whenever Pk is true
Hence, by the principle of mathematical induction, Pn is true for all n∈N

By using PMI, Prove that, x2n- y2nis divisible by x+y.

By using PMI,
Prove that, x²ⁿ- y²ⁿis divisible by x+y.