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: 58

Answer

$S_{23}=310.5$

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: $a=-3 \\a_n = 30 \\d=\frac{3}{2}-0=\frac{3}{2}$ 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: $\require{cancel} a_n = a + (n-1)d \\30 = -3+(n-1)\frac{3}{2} \\30+3 = (n-1)(\frac{3}{2}) \\33=(n-1)(\frac{3}{2}) \\\frac{2}{3}(33)=(n-1)(\frac{3}{2}) \cdot \frac{2}{3} \\\frac{2}{\cancel{3}}(\cancel{3}(11))=(n-1)(\frac{\cancel{3}}{\cancel{2}}) \cdot \frac{\cancel{2}}{\cancel{3}}\\ \\22=n-1 \\22+1=n-1+1 \\23=n$ Now that it is known that $n=23$, the sum of the first 23 terms can be computed using the formula above. $S_{23} = \dfrac{23}{2}(-3+30) \\S_{23}=\dfrac{23}{2}(27) \\S_{23}=310.5$
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