Question

5. 14 marks In this problem we will explore the factors of expression a. 12 marks] Suppose a friend comes to you claiming that "(s ) divides sWrite your friend's claim as a theorem, specifying the domains for each of the variables and using appropriate quantifiers. b. 5 marks Use mathematical induction to prove the theorem! Hint: sk-can be rewritten in the form (s-11) + z where r is an expression involving s, t, k and some constants tk-1) +1 where x is an expression involving s, t, k and some constants n the form SS

Answer 1

We are presenting a consolidated answer here:

Let the given statement be P(n). Then we need to prove that:

P(k): (sⁿ - tⁿ) is divisible by (s - t).

When n = 1, the given statement becomes: (s¹ - t¹) is divisible by (s - t), which stands true, hence P(1) is true.

Let p(k) be true. Then P(k): s^k - t^k is divisible by (s-t).

Now, s^{k + 1} - t^{k + 1} = s^{k + 1} - s^kt + s^kt - t^{k + 1} [on adding and subtracting s^kt]

= s^k(s - t) + t(s^k - t^k), which is divisible by (s - t) [using (i)]

⇒ P(k + 1): s^{k + 1} - t^{k + 1}is divisible by (s - t)

⇒ P(k + 1) is true, whenever P(k) is true.

Thus as P(1) and P(k + 1) stand true, whenever P(k) is true. Hence, by the Principal of Mathematical Induction, P(n) is true for all n ∈ N.