# Chapter 5 - Number Theory and the Real Number System - 5.7 Arithmetic and Geometric Sequences - Exercise Set 5.7 - Page 330: 94

General Term Formula: $a_n = 12 \cdot \left(\frac{1}{2}\right)^{n-1}$ $a_7 = \frac{3}{16}$

#### Work Step by Step

RECALL: 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: $a_1 = 12$ $r = \frac{6}{12} =\frac{1}{2}$ (2) Substitute the values of $a_1$ and $r$ in the formula above. Substituting gives us: $a_n = 12 \cdot \left(\frac{1}{2}\right)^{n-1}$ Therefore, the 7th term of the sequence is: $a_{7}=12 \cdot \left(\frac{1}{2}\right)^{7-1} \\a_{7}=12 \cdot \left(\frac{1}{2}\right)^6 \\a_{7} = 12 \cdot \frac{1}{64} \\a_{7} = \frac{12}{64} \\a_{7}=\frac{3}{16}$

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