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: 62

Answer

$S_{100}=-505$

Work Step by Step

RECALL: (1) The sum of the first $n$ terms of an arithmetic sequence is given by the formula: $S_n=\dfrac{n}{2}(a+a_n)$ where $a$ = first term $d$ = common difference $a_n$ = $n^{th}$ term (2) The $n^{th}$ term $a_n$ of an arithmetic sequence is given by the formula: $a_n = a + (n-1)d$ where $a$ = first term $d$ = common difference The given arithmetic sequence has: $a=-10 \\a_n = -0.1 \\d=-9.9-(-10)=-9.9+10=0.1$ The formula for the partial sum requires the values of $a$, $a_n$ and $n$. However, only $a$ and $a_n$ are known at the moment. Solve for $n$ using the formula for $a_n$ to obtain: $\require{cancel} a_n = a + (n-1)d \\-0.1 = -10+(n-1)(0.1) \\-0.1+10 = (n-1)(0.1) \\9.9=(n-1)(0.1) \dfrac{9.9}{0.1}=\dfrac{(n-1)(0.1)}{0.1} \\99=n-1 \\99+1=n-1+1 \\100=n$ Now that it is known that $n=100$, the sum of the first 100 terms can be computed using the formula above. $S_{100} = \dfrac{100}{2}[-10+(-0.1)] \\S_{100}=50(-10.1) \\S_{100}=-505$
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