# Chapter 9 - Section 9.3 - Geometric Sequences; Geometric Series - 9.3 Assess Your Understanding - Page 664: 27

$a_7=\dfrac{1}{64}$

#### 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 $a_1=1$. Solve for the common ratio using the formula in (2) above to obtain: $r = \dfrac{a_2}{a_1}=\dfrac{\frac{1}{2}}{1}=\dfrac{1}{2}$ Thus, the $n^{th}$ term of the sequence is given by the formula: $a_n = 1 \cdot (\frac{1}{2})^{n-1}$ The 7th term can be found by substituting $7$ for $n$: $a_7=1 \cdot (\frac{1}{2})^{7-1} \\a_7=1 \cdot (\frac{1}{2})^6 \\a_7 = 1 \cdot \dfrac{1}{64} \\a_7=\dfrac{1}{64}$

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