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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