Algebra and Trigonometry 10th Edition

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

Chapter 11 - 11.4 - Mathematical Induction - 11.4 Exercises - Page 806: 17


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 $n=1$: $\frac{1(1+1)}{2}=\frac{1(2)}{2}=1$ It is correct! Now, suppose that the formula is correct, that is: $1+2+3+4+...+n=\frac{n(n+1)}{2}$ Now, let's prove the formula for $n+1$: $1+2+3+4+...+n+(n+1)=(1+2+3+4+...+n)+(n+1)=\frac{n(n+1)}{2}+(n+1)=\frac{n(n+1)}{2}+\frac{2(n+1)}{2}=\frac{(n+1)(n+2)}{2}=\frac{(n+1)[(n+1)+1]}{2}$ That is exactly the given formula if $n$ is changed by $n+1$
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