Algebra and Trigonometry 10th Edition

Published by Cengage Learning
ISBN 10: 9781337271172
ISBN 13: 978-1-33727-117-2

Chapter 11 - Review Exercises - Page 841: 65


The formula was proved for $n=1$ The formula is correct if $n$ is changed by $n+1$

Work Step by Step

Let's prove the formula for $n=1$: $1(1+2)=1(3)=3$ It is correct! Now, suppose that the formula is correct, that is: $3+5+7+...+2n+1=n(n+2)$ Now, let's prove the formula for $n+1$: $3+5+7+...+2n+1+2(n+1)+1=(3+5+7+...+2n+1)+2n+2+1=n(n+2)+2n+3=n^2+2n+2n+3=n^2+n+3n+3=n(n+1)+3(n+1)=(n+1)(n+3)=(n+1)[(n+1)+2]$ That is exactly the given formula if $n$ is changed by $n+1$
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