Prove the statement by the Principle of Mathematical Induction :For a

1.	Prove the statement by the Principle of Mathematical Induction :For any natural number n, xn – yn is divisible by x – y, where x integers with x ≠ y.
Answer» According to the question, P(n) = xⁿ – yⁿ is divisible by x – y, x integers with x ≠ y. So, substituting different values for n, we get, P(0) = x⁰ – y⁰ = 0 Which is divisible by x − y. P(1) = x − y Which is divisible by x − y. P(2) = x² – y² = (x +y)(x−y) Which is divisible by x−y. P(3) = x³ – y³ = (x−y)(x²+xy+y²) Which is divisible by x−y. Let P(k) = x^k – y^k be divisible by x – y; So, we get, ⇒ x^k – y^k = a(x−y). Now, we also get that, ⇒ P(k+1) = x^k+1 – y^k+1 = x^k(x−y) + y(x^k−y^k) = x^k(x−y) +y a(x−y) Which is divisible by x − y. ⇒ P(k+1) is true when P(k) is true. Therefore, by Mathematical Induction, P(n) xⁿ – yⁿ is divisible by x – y, where x integers with x ≠ y which is true for any natural number n.

Prove the statement by the Principle of Mathematical Induction :For any natural number n, xn – yn is divisible by x – y, where x integers with x ≠ y.

Answer»

According to the question,

P(n) = xⁿ – yⁿ is divisible by x – y, x integers with x ≠ y.

So, substituting different values for n, we get,

P(0) = x⁰ – y⁰ = 0 Which is divisible by x − y.

P(1) = x − y Which is divisible by x − y.

P(2) = x² – y²

= (x +y)(x−y) Which is divisible by x−y.

P(3) = x³ – y³

= (x−y)(x²+xy+y²) Which is divisible by x−y.

Let P(k) = x^k – y^k be divisible by x – y;

So, we get,

⇒ x^k – y^k = a(x−y).

Now, we also get that,

⇒ P(k+1) = x^k+1 – y^k+1

= x^k(x−y) + y(x^k−y^k)

= x^k(x−y) +y a(x−y) Which is divisible by x − y.

⇒ P(k+1) is true when P(k) is true.

Therefore, by Mathematical Induction,

P(n) xⁿ – yⁿ is divisible by x – y, where x integers with x ≠ y which is true for any natural number n.