Geometric Sequence using Logarithms

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 \(10 \ 000\).

\( \begin{aligned} \displaystyle
1000 &\lt 2^{n-1} \lt 10000 \\
\log_{2}{1000} &\lt n-1 \lt \log_{2}{10000} \\
1+ \log_{2}{1000} &\lt n \lt 1+ \log_{2}{10000} \\
10.965 \cdots &\lt n \lt 14.287 \cdots \\
11 &\le n \le 14 \\
n &= 11, 12, 13, 14 \\
\therefore \text{There are 4 terms.}
\end{aligned} \)

Unlock your full learning potential—download our expertly crafted slide files for free and transform your self-study sessions!

Discover more enlightening videos by visiting our YouTube channel!

 

Algebra Algebraic Fractions Arc Binomial Expansion Capacity Common Difference Common Ratio Differentiation Double-Angle Formula Equation Exponent Exponential Function Factorise Functions Geometric Sequence Geometric Series Index Laws Inequality Integration Kinematics Length Conversion Logarithm Logarithmic Functions Mass Conversion Mathematical Induction Measurement Perfect Square Perimeter Prime Factorisation Probability Product Rule Proof Pythagoras Theorem Quadratic Quadratic Factorise Ratio Rational Functions Sequence Sketching Graphs Surds Time Transformation Trigonometric Functions Trigonometric Properties Volume




Related Articles

Responses

Your email address will not be published. Required fields are marked *