Applications of Geometric Sequence using Logarithms
Geometric Sequence using Logarithms is used to find the number of geometric sequences terms.
Question 1
Find the largest \(5\)-digit term of the sequence \( 2, 4, 8, 16, \cdots \).
\( \begin{aligned} \displaystyle
2 \times 2^{n-1} &\lt 100000 \\
2^{n} &\lt 100000 \\
n &\lt \log_{2}{100000} \\
n &\lt 16.609 \cdots \\
n &= 16 \\
\therefore 2^{16} &= 65536
\end{aligned} \)
Question 2
Find the smallest 5 digits term of the sequence \( 1, 3, 9, 27, \cdots \).
\( \begin{aligned} \displaystyle
3^{n-1} &\gt 10000 \\
n-1 &\gt \log_{3}{10000} \\
n &\gt \log_{3}{10000} + 1 \\
n &\gt 9.38 \cdots \\
n &= 10 \\
\therefore 3^{10-1} &= 59049
\end{aligned} \)
Question 3
Find the number of terms in the sequence \( 1, 2, 4, 8, \cdots \) between \(1000\) and \(10 \ 000\).
\( \begin{aligned} \displaystyle
1000 &\lt 2^{n-1} \lt 10000 \\
\log_{2}{1000} &\lt n-1 \lt \log_{2}{10000} \\
1+ \log_{2}{1000} &\lt n \lt 1+ \log_{2}{10000} \\
10.965 \cdots &\lt n \lt 14.287 \cdots \\
11 &\le n \le 14 \\
n &= 11, 12, 13, 14 \\
\therefore \text{There are 4 terms.}
\end{aligned} \)
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