College Algebra 7th Edition

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

Chapter 8, Sequences and Series - Chapter 8 Review - Exercises - Page 640: 51

Answer

Arithmetic; $5050\sqrt 5$

Work Step by Step

Let's note by $\{a_n\}$ the given sequence. The sum of its first $k$ terms is: $$S_k=\sqrt 5+2\sqrt 5+3\sqrt 5+\dots+100\sqrt 5.$$ The terms of the sequence are: $$\begin{align*} a_1&=\sqrt 5\\ a_2&=2\sqrt 5\\ a_3&=3\sqrt 5\\ &\dots\\ a_k&=100\sqrt 5. \end{align*}$$ We notice that the common difference of consecutive terms is $2\sqrt 5-\sqrt 5=3\sqrt 5-2\sqrt 5=\sqrt 5$, therefore constant, so the sequence is arithmetic with the elements: $$\begin{cases} a_1=\sqrt 5\\ d=\sqrt 5. \end{cases}$$ Before calculating $S_k$ we must find $k$ (the number of terms in the sum): $$\begin{align*} a_k&=a_1+(k-1)d\\ 100\sqrt 5&=\sqrt 5+(k-1)\sqrt 5\\ 100&=1+k-1\\ k&=100. \end{align*}$$ Calculate the partial sum $S_{100}$ using the formula: $$S_n=\dfrac{n(a_1+a_n)}{2}.$$ For $n=100$ we have: $$S_{100}=\dfrac{100(\sqrt 5+100\sqrt 5)}{2}=5050\sqrt 5.$$
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