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Chapter 12: Q 25. (page 830)

In Problem, prove each statement.

a-bis a factor ofan-bn

Short Answer

Expert verified

The given statement is shown.

Step by step solution

01

Step 1. Given information

The expression isa+b.

02

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

Put n=1in an-bn

a1-b1=a-b

a-bis a factor of a-b

Thus, the statement is true for n=1

Assume that the statement is true for n=k

Thus, a-bis a factor of ak-bk.

Find whether a-bis a factor of ak+1-bk+1

width="190">ak+1-bk+1=ak路a-bk路b

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

a-bis a factor of ak-bk. Thus, a-bis a factor of dk+1-bk+1.

Thus, the statement is true fork+1

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

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