   Chapter 2.2, Problem 10E

Chapter
Section
Textbook Problem

Prove that the statements in Exercises 1 − 16 are true for every positive integer n . 1 ⋅ 2 + 2 ⋅ 2 2 + 3 ⋅ 2 3 + ⋅ ⋅ ⋅ + n ⋅ 2 n = ( n − 1 ) 2 n + 1 + 2

To determine

To prove: That 12+222+323++n2n=(n1)2n+1+2 is true for all n by using mathematical induction.

Explanation

Formula Used:

Considering the given statement is Pn for all integers n,

a. if Pn is true for n=1

b. if the truth of Pk always implies that Pk+1 is true, then the statement Pn is true for all positive integers n.

Proof:

For each positive integer n, let Pn be the statement

12+222+323++n2n=(n1)2n+1+2

Here, n2n is the last term on the left side.

When n=1, there is only one term, and no addition is actually performed.

When n=1, the value of the left side is

(1)21=2

And the value of right side is

(01)21+1+2=0+2=2

Thus P1 is true.

Assume now that Pk is true. That is, assume that the equation

12+222+323++k2k=(k1)2k+1+2 is true.

With this assumption made, prove that Pk+1 is true. Add (k+1)2k+1 to both sides of the assumed equality, the following result is obtained

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