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Chapter 12: Q. 9 (page 830)

In Problems 1鈥�22, use the Principle of Mathematical Induction to show that the given statement is true for all natural numbers n.

1+4+42+...+4n-1=134n-1

Short Answer

Expert verified

The statement1+4+42+...+4n-1=134n-1is true for all natural numbers.

Step by step solution

01

Step 1. Given information

The given statement is 1+4+42+...+4n-1=134n-1.We need to prove that the given statement is true for all natural numbers by using mathematical induction.

02

Step 2. Proof

Let's prove that the statement is true for n=1.

41-1=1341-1.

40=133.

1=1.

So, the statement is true for n=1.

Let's assume that the statement is true for n=k.

So, role="math" localid="1648054611339" 1+4+42+...+4k-1=134k-1.

Let's prove that the statement is true for n=k+1.

1+4+42+...+4k-1+4k+1-1.

=134k-1+4k.

=43路4k-13.

=134k+1-1.

The statement is true for n=k+1.

Since the given statement is true for n=1and if the statement is true for some n=kand it is also true for the next natural numberk+1.

So, the statement is true for all natural numbers.

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