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Elements Of Modern Algebra

8th Edition
Gilbert + 2 others
ISBN: 9781285463230

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Section
BuyFindarrow_forward

Elements Of Modern Algebra

8th Edition
Gilbert + 2 others
ISBN: 9781285463230
Textbook Problem

Show that if the statement

1 + 2 + 3 + ... + n = n ( n + 1 ) 2 + 2

is assumed to be true for n = k , the same equation can be proved to be true for n = k + 1 . Explain why this does not prove that the statement is true for all positive integers. Is the statement true for all positive integers? Why?

To determine

Whether the statement “ 1+2+3++n=n(n+1)2+2 is assumed to be true for n=k,” then it is true for n=k+1 and explain why the statement is not true for all positive integers n.

Explanation

Given information:

The given statement is, “ 1+2+3++n=n(n+1)2+2 ”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+3++n=n(n+1)2+2

Assume that Pk is true.

1+2+3++k=k(k+1)2+2

For n=k+1

By induction hypothesis, 1+2+3++k=k(k+1)2+2

Adding k+1 on both sides,

1+2+3++k+(k+1)=k(k+1)2+2+(k+1)

Simplify R

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