Precalculus: Mathematics for Calculus, 7th Edition

Published by Brooks Cole
ISBN 10: 1305071751
ISBN 13: 978-1-30507-175-9

Chapter 12 - Section 12.2 - Arithmetic Sequences - 12.2 Exercises - Page 857: 30


Sequence is arithmetic $d=\frac{1}{2}$ $\ a_{n}=\dfrac {3}{2}+\dfrac {1}{2}\left( n-1\right) $

Work Step by Step

In order a sequence to be arithmetic $a_{n+1}-a_{n}=a_{n}-a_{n-1}\Rightarrow 2a_{n}=a_{n+1}+a_{n-1}$ $2a_{n}=2\left( 1+\dfrac {n}{2}\right) =2+n$ $a_{n+1}+a_{n-1}=1+\dfrac {n+1}{2}+1+\dfrac {n-1}{2}=2+n$ $\Rightarrow 2a_{n}=a_{n+1}+a_{n-1}$ So this sequence is arithmetic $a_{n}=1+\dfrac {n}{2}=a+\left( n-1\right) d=a-d+nd\Rightarrow d=\dfrac {1}{2};a-d=1\Rightarrow a=\dfrac {3}{2}$ $\Rightarrow a_{n}=a+\left( n-1\right) d=\dfrac {3}{2}+\dfrac {1}{2}\left( n-1\right) $
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