# Geometric Sequence using Logarithms

## 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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