Question

Use mathematical induction to prove that \(a - b\)is a factor

of \({a^n} - {b^n}\)whenever \(n\)is a positive integer.

Short answer

It is proved that\(a - b\)is a factor of\({a^n} - {b^n}\)whenever\(n\)is a positive integer.

Step-by-step solution

Principle of Mathematical Induction

Consider the propositional function \(P\left( n \right)\). Consider two actions to prove that\(P\left( n \right)\)evaluates to accurate for all set of positive integers\(n\).

Consider the first basic step is to confirm that \(P\left( 1 \right)\) true.

Consider the inductive step is to demonstrate that for any positive integer k the conditional statement\(P\left( k \right) \to P\left( {k + 1} \right)\)is true.

Prove the basis step

Let\(P\left( n \right)\): “\(a - b\)is a factor of\({a^n} - {b^n}\)”

In the basis step, one needs to prove that\(P\left( 1 \right)\)is true

For finding statement\(P\left( 1 \right)\)substituting\(1\)for\(n\)in the statement solve as:

\({a^n} - {b^n} = {a^1} - {b^1} = a - b\)

From the above, one can see that the statement \(P\left( 1 \right)\) is true this is also known as the basis step of the proof.

Prove the Inductive step

In the inductive step, we need to prove that, if\(P\left( k \right)\)is true, then\(P\left( {k + 1} \right)\)is also true.

Consider\(P\left( k \right) \to P\left( {k + 1} \right)\)is true for all positive integers k.

In the inductive hypothesis, we assume that\(P\left( k \right)\)is true for any arbitrary positive integer\(k\). Then,\(a - b\)is a factor of\({a^k} - {b^k}\)

Now one have to show that\(P\left( {k + 1} \right)\)is also true

Therefore, replacing\(k\)with\(k + 1\)in the statement

\(\begin{array}{c}{a^{k + 1}} - {b^{k + 1}} = a{a^k} - b{b^k}\\ = a{a^k} - {a^k}b + {a^k}b - b{b^k}\\ = {a^k}\left( {a - b} \right) + b\left( {{a^k} - {b^k}} \right)\end{array}\)

The first term of the expression has a factor\(a - b\)and the second terms also have a factor of\(a - b\)from the inductive hypothesis.

Since both the terms have a factor of\(a - b\), therefore\({a^{k + 1}} - {b^{k + 1}}\)have a factor of \(a - b\).

From the above, it is seen that\(P\left( {k + 1} \right)\)is also true.

Hence,\(P\left( {k + 1} \right)\)is true under the assumption that\(P\left( k \right)\)is true. This

completes the inductive step.

Hence, It is proved that\(a - b\)is a factor of\({a^n} - {b^n}\)whenever nis a positive integer.