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Chapter 12: Q 17. (page 830)

In Problem, use the Principle of Mathematical Induction to show that the given statement is true for all natural numbers n.

role="math" localid="1647675344162" 1路2+2路3+3路4+...+n(n+1)=13n(n+1)(n+2)

Short Answer

Expert verified

The given statement has been shown.

Step by step solution

01

Step 1. Given information

Statement is1路2+2路3+3路4+...+n(n+1)=13n(n+1)(n+2)

02

Step 2. Show the given statement using the principle of mathematical induction.

For n=1

13(1)(1+1)(1+2)=13(2)(3)=2

Assume that the statement is true for n=k.

1路2+2路3+3路4+鈥�鈥�鈥�鈥�+k(k+1)=13k(k+1)(k+2)

style="max-width: none; vertical-align: -89px;" width="552" 13(k+1)((k+1)+1)((k+1)+2)=1路2+2路3+鈥�鈥�鈥�+k(k+1)+(k+1)((k+1)+1)=13k(k+1)(k+2)+(k+1)(k+2)=13(k+1)(k+1+1)[k+1+2]=13(k+1)[(k+1)+1][(k+1)+2]

Thus, the statement is true for k + 1.

Since both the conditions of the Principle of Mathematical Induction are satisfied, the statement is true for all natural numbers.

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