Question

How many terms are needed in the following geometric sequence to get within 10^-9 of zero? b.) i.) 4, -2, 1

Answer 1

$$
\Large a_n = 4(-\frac{1}{2})^{n-1}

$$

Answer 2

so you need to solve
$$
\Large {|4(- \frac{1}{2} ) ^{n-1} - 0 | \leq 10^{-9}
}
$$

Answer 3

$$
\Large {
|4(- \frac{1}{2} ) ^{n-1} - 0 | \leq 10^{-9}

\\ \therefore \\
4( \frac{1}{2} ) ^{n-1} \leq 10^{-9}
\\ ~ \\
( \frac{1}{2} ) ^{n-1} \leq \frac{10^{-9} }{4}
\\~\\
\log ( \frac{1}{2} ) ^{n-1} \leq \log (\frac{10^{-9} }{4})
\\~\\
(n-1) \log (\frac{1}{2}) \leq \log (\frac{10^{-9} }{4})
\\~\\
(n-1) \leq \frac{\log (\frac{10^{-9} }{4})}{ \log (\frac{1}{2})}
\\~\\
n \leq \frac{\log (\frac{10^{-9} }{4})}{ \log (\frac{1}{2})}+1


}

$$