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: 38

Answer

$a_n=21\cdot (\frac{1}{3})^{n-1}$

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: $r = \dfrac{a_n}{a_{n-1}}$ Note that: $r=\dfrac{a_2}{a_1}$ The given geometric sequence has: $a_2=7$ $r=\frac{1}{3}$ Substitute these values into the equation involving $r$ above to obtain: $r=\dfrac{a_2}{a_1} \\\dfrac{1}{3} = \dfrac{7}{a_1}$ Cross-multiply to obtain: $1(a_1)=3(7) \\a_1=21$ Therefore, the $n^{th}$ term of the sequence is given by the formula: $a_n= a_1 \cdot r^{n-1} \\a_n=21\cdot (\frac{1}{3})^{n-1}$
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