Question

In Problem, prove each statement.

$a-b$is a factor of$a^n-b^n$

Short answer

The given statement is shown.

Step-by-step solution

Step 1. Given information

The expression is$a+b$.

Step 2. Prove that a-b is a factor of an-bn.

Put $n=1$in $a^n-b^n$

$a^1-b^1=a-b$

$a-b$is a factor of $a-b$

Thus, the statement is true for $n=1$

Assume that the statement is true for $n=k$

Thus, $a-b$is a factor of $a^k-b^k$.

Find whether $a-b$is a factor of $a^{k+1}-b^{k+1}$

width="190">ak+1-bk+1=ak·a-bk·b

width="352">ak·a-bk·b+abk-abk=ak·a-abk+abk-bk·b=aak-bk+bk(a-b)

$a-b$is a factor of $a^k-b^k$. Thus, $a-b$is a factor of $d^{k+1}-b^{k+1}$.

Thus, the statement is true for$k+1$

Since both the conditions of the Principle of Mathematical Induction are satisfied, the statement