Thinking Mathematically (6th Edition)

Published by Pearson
ISBN 10: 0321867327
ISBN 13: 978-0-32186-732-2

Chapter 5 - Number Theory and the Real Number System - 5.7 Arithmetic and Geometric Sequences - Exercise Set 5.7: 91

Answer

General Term Formula: $a_n = 3 \cdot 4^{n-1}$ $a_7=12,288$

Work Step by Step

The formula for the general term (nth term) of a geometric sequence is: $a_n=a_1 \cdot r^{n-1}$ where r = common ratio $a_1$ = first term $a_n$ = nth term $n$= term number To find the formula for the general term of the given geometric sequence, perform the following steps: (1) Solve/find the values of $a_1$ and $r$ The given sequence has: $a_1 = 3$ $r = \frac{12}{3} =4$ (2) Substitute the values of $a_1$ and $r$ in the formula above. Substituting gives us: $a_n = 3 \cdot 4^{n-1}$ Therefore, the 7th term of the sequence is: $a_{7}=3 \cdot 4^{7-1} \\a_{7}=3 \cdot 4^6 \\a_{7} = 3 \cdot 4096 \\a_{7} = 12,288$
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