   Chapter 2.2, Problem 49E

Chapter
Section
Textbook Problem

Show that if the statement 1 + 2 + 2 2 +   ... + 2 n − 1 = 2 n is assumed to be true for n = k , then it can be proved to be true for n = k + 1 . Is the statement true for all positive integers n ? Why?

To determine

Whether the statement,“ 1+2+22++2n1=2n is assumed to be true for n=k,” then it is true for n=k+1 and verify the statement is true for all positive integers n or not.

Explanation

Given information:

The given statement is, “ 1+2+22++2n1=2n ”is true for n=k.

Formula used:

For all positive integers n, the statement Pn is true if,

a. Pn is true for n=1

b. The truth of Pk always implies that Pk+1 is true.

Proof:

Let Pn be the statement, “ 1+2+22++2n1=2n

Assume that Pk is true.

1+2+22++2k1=2k

For n=k+1

By induction hypothesis, 1+2+22+

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