College Algebra 7th Edition

Published by Brooks Cole
ISBN 10: 1305115546
ISBN 13: 978-1-30511-554-5

Chapter 8, Sequences and Series - Section 8.2 - Arithmetic Sequences - 8.2 Exercises: 41

Answer

$d=1.5$ $a_5=31$ The $n^{th}$ term is given by: $a_n = 25+1.5(n-1)$ $a_{100} = 173.5$

Work Step by Step

The sequence is arithmetic so the terms have a common difference. The common difference $d$ can be found by subtracting any term to the next term in the sequence. Thus, $d=26.5-25 \\d=1.5$ The fifth term $a_5$ can be found by adding the common difference $1.5$ to the fourth term. The fourth term of the sequence is $29.5$. Thus, $a_5 = 29.5+1.5 \\a_5=31$ The $n^{th}$ term $a_n$ of an arithmetic sequence is given by the formula $a_n = a+d(n-1)$ where $a$ = first term and $d$ = common difference. The sequence has $a=25$ and $d=1.5$. Thus, the $n^{th}$ term is given by: $a_n = 25+1.5(n-1)$ Substituting 100 to $n$ gives the 100th term as: $a_{100} = 25+1.5(100-1) \\a_{100} = 25+1.5(99) \\a_{100} = 25+148.5 \\a_{100} = 173.5$
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