Precalculus (10th Edition)

Published by Pearson
ISBN 10: 0-32197-907-9
ISBN 13: 978-0-32197-907-0

Chapter 12 - Sequences; Induction; the Binomial Theorem - 12.2 Arithmetic Sequences - 12.2 Assess Your Understanding - Page 814: 49



Work Step by Step

We have to determine the sum: $S=4+4.5+5+5.5+...+100$ As $4.5-4=5-4.5=5.5-5=...=0.5$, the sequence is arithmetic. Determine its first term and the common difference: $a_1=4$ $d=0.5$ Determine the number of terms: $a_n=a_1+(n-1)d$ $a_n-a_1=(n-1)d$ $n-1=\dfrac{a_n-a_1}{d}$ $n=\dfrac{a_n-a_1}{d}+1$ $n=\dfrac{100-4}{0.5}+1$ $n=193$ Therefore, the given sum contains $193$ terms, so we have to determine the sum of the first $193$ terms. We use the formula: $S_n=\dfrac{n(a_1+a_n)}{2}$ $4+4.5+5+5.5+...+100=\dfrac{193(100+4)}{2}$ $=\dfrac{193\cdot 104}{2}$ $=10,036$
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