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: 16


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