College Algebra 7th Edition

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

Chapter 8, Sequences and Series - Section 8.3 - Geometric Sequences - 8.3 Exercises: 46

Answer

$a=648$ $a_7=\dfrac{1}{72}$

Work Step by Step

To find the first three terms, the value of the common ratio $r$ is needed. Note that the previous term of a geometric sequence can be found by dividing the common ratio $r$ to the current term. This means that ti find the value of the first term, the third term must be divided by the common ratio twice (or by $r^2$). The geometric sequence has: $r=\frac{1}{6}$ $a_3=18$ To find the first term, divide $a_3$ by the common ratio $r$ twice (or by $r^2$) to obtain: $a = \dfrac{a_3}{r^2} \\a=\dfrac{18}{(\frac{1}{6})^2} \\a=\dfrac{18}{\frac{1}{36}} \\a=18 \cdot \dfrac{36}{1} \\a= 648$ The $n^{th}$ term of a geometric sequence can be found using the formula $a_n=a \cdot r^{n-1}$ where $a$ = first term and $r$ = common ratio. Thus, the $n^{th}$ term of the given sequence is given by the formula: $a_n = a \cdot r^{n-1} \\a_n = 648 \cdot \left(\dfrac{1}{6}\right)^{n-1}$ Thus, the 7th term of the sequence is: $a_7 = 648 \cdot \left(\dfrac{1}{6}\right)^{7-1} \\a_7 = 648 \cdot \left(\dfrac{1}{6}\right)^6 \\a_7=\dfrac{1}{72}$
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