College Algebra (10th Edition)

Published by Pearson
ISBN 10: 0321979478
ISBN 13: 978-0-32197-947-6

Chapter 9 - Section 9.3 - Geometric Sequences; Geometric Series - 9.3 Assess Your Understanding: 37

Answer

$a_n=-1 \cdot (-3)^{n-1}$

Work Step by Step

RECALL: (1) The $n^{th}$ term $a_n$ of a geometric sequence is given by the formula $a_n=a_1 \cdot r^{n-1}$ where $a_1$ = first term $r$ = common ratio (2) The common ratio of a geometric sequence is equal to the quotient of any term and the term before it: $r = \dfrac{a_n}{a_{n-1}}$ The given geometric sequence has: $a_6=243$ $r=-3$ Note that: $a_6=a_1 \cdot (r)^{6-1} \\a_6=a_1 \cdot r^5$ Substitute the given values f $a_6$ and $r$ to obtain: $a_6 = a_1 \cdot r^5 \\243=a_1 \cdot (-3)^5 \\243=a_1 \cdot (-243) \\\dfrac{243}{-243} = a_1 \\-1=a_1$ Thus, the $n^{th}$ term of the sequence is given by the formula: $a_n = a_1 \cdot r^{n-1} \\a_n=-1 \cdot (-3)^{n-1}$
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