# Nakafa Learning Content > For AI agents: use [llms.txt](https://nakafa.com/llms.txt) for the site index. Markdown versions are available by appending `.md` to content URLs or sending `Accept: text/markdown`. URL: https://nakafa.com/en/subjects/mathematics/sequence-series/geometric-sequence Source: https://raw.githubusercontent.com/nakafaai/nakafa.com/refs/heads/main/packages/contents/material/lesson/mathematics/sequence-series/geometric-sequence/en.mdx Learn geometric sequences with constant ratios. Learn formulas, solve real-world problems like bacterial growth, and discover practical applications. --- ## Understanding Geometric Sequences A geometric sequence is a sequence of numbers in which each term has a constant ratio to the previous term. This ratio is denoted by the letter $$r$$. Visible text: A geometric sequence is a sequence of numbers in which each term has a constant ratio to the previous term. This ratio is denoted by the letter . If we have a geometric sequence $$U_1, U_2, U_3, ..., U_n$$, then: Visible text: If we have a geometric sequence , then: ```math \frac{U_2}{U_1} = \frac{U_3}{U_2} = \frac{U_4}{U_3} = ... = r ``` In other words, each term in a geometric sequence is obtained by multiplying the previous term by the ratio $$r$$. Visible text: In other words, each term in a geometric sequence is obtained by multiplying the previous term by the ratio . ## Paper Folding Exploration Let's perform a simple exploration to understand the concept of a geometric sequence. Prepare a rectangular piece of paper and fold it several times. If the paper is folded once, it will be divided into $$2$$ equal parts. If folded again (twice), it will form $$4$$ equal parts. The following pattern emerges: Visible text: If the paper is folded once, it will be divided into equal parts. If folded again (twice), it will form equal parts. The following pattern emerges: | Number of folds | Number of equal parts | | --------------- | --------------------- | | $$1 \text{ fold}$$ | $$2 \text{ parts}$$ | | $$2 \text{ folds}$$ | $$4 \text{ parts}$$ | | $$3 \text{ folds}$$ | $$8 \text{ parts}$$ | | $$4 \text{ folds}$$ | $$16 \text{ parts}$$ | Visible text: | Number of folds | Number of equal parts | | --------------- | --------------------- | | | | | | | | | | | | | Notice that the number of parts formed creates a sequence: $$2, 4, 8, 16, \ldots$$ Visible text: Notice that the number of parts formed creates a sequence: In this sequence, each term is obtained by multiplying the previous term by $$2$$. In other words, the ratio is $$2$$. Visible text: In this sequence, each term is obtained by multiplying the previous term by . In other words, the ratio is . ## General Formula for Geometric Sequences The general formula for a geometric sequence is: ```math U_n = a \cdot r^{n-1} ``` Where: - $$U_n$$ = general term - $$a$$ = first term - $$r$$ = ratio - $$n$$ = term number Visible text: - = general term - = first term - = ratio - = term number ## Bacterial Division Bacteria reproduce by dividing themselves. Within two hours, one bacterial cell divides into $$3 \text{ parts}$$. Visible text: Bacteria reproduce by dividing themselves. Within two hours, one bacterial cell divides into . If the initial number of bacteria is $$2 \text{ cells}$$, then: Visible text: If the initial number of bacteria is , then: - First term ($$U_1$$) $$= 2$$ - Ratio ($$r$$) $$= 3$$ Visible text: - First term () - Ratio () In $$20 \text{ hours}$$, division occurs $$10\text{ times}$$ ($$20 \div 2 = 10$$). Visible text: In , division occurs (). To determine the number of bacteria after $$20 \text{ hours}$$ ($$10$$th term), we use the formula: Visible text: To determine the number of bacteria after (th term), we use the formula: ```math U_{10} = 2 \cdot 3^{10-1} = 2 \cdot 3^9 = 2 \cdot 19{,}683 = 39{,}366 ``` So, after $$20 \text{ hours}$$, there are $$39{,}366 \text{ bacterial cells}$$. Visible text: So, after , there are . Component: BacterialGrowth Props: - formulaType: geometric - ratio: 3 - initialCount: 2 - maxGenerations: 3 - timeInterval: 2 - labels: { title: "Bacterial Division", bacterial: "Bacteria", initialBacteria: "Initial Bacteria", } ## Properties of Geometric Sequences ### Ratio in Geometric Sequences The ratio ($$r$$) in a geometric sequence is always constant and can be calculated by dividing the next term by the previous term: Visible text: The ratio () in a geometric sequence is always constant and can be calculated by dividing the next term by the previous term: ```math r = \frac{U_2}{U_1} = \frac{U_3}{U_2} = \frac{U_4}{U_3} = ... ``` ### Finding the nth Term To find the general term of a geometric sequence, we can use the formula: ```math U_n = a \cdot r^{n-1} ``` ## Applications of Geometric Sequences Geometric sequences are applied in various fields, such as: 1. Population growth (as in the bacteria example) 2. Compound interest in economics 3. Radioactive decay in physics 4. Cell growth in biology By understanding the concept of geometric sequences, we can model and predict various phenomena involving growth or decrease with a constant ratio. ## Example Problems ### Finding the Ratio The first term of a geometric sequence is $$4$$ and the fourth term is $$108$$. Determine the ratio of this sequence. Visible text: The first term of a geometric sequence is and the fourth term is . Determine the ratio of this sequence. **Solution**: Given: - $$a = 4$$ (first term) - $$U_4 = 108$$ (fourth term) Visible text: - (first term) - (fourth term) Using the general formula for geometric sequences: Component: MathContainer Children: ```math U_4 = a \cdot r^{4-1} ``` ```math 108 = 4 \cdot r^3 ``` ```math r^3 = \frac{108}{4} = 27 ``` ```math r = \sqrt[3]{27} = 3 ``` Therefore, the ratio of the geometric sequence is $$3$$. Visible text: Therefore, the ratio of the geometric sequence is . ### Length of Rope Sections A rope is divided into $$5 \text{ parts}$$ with lengths forming a geometric sequence. If the shortest piece is $$16 \text{ cm}$$ and the longest piece is $$81 \text{ cm}$$, determine the length of the third piece. Visible text: A rope is divided into with lengths forming a geometric sequence. If the shortest piece is and the longest piece is , determine the length of the third piece. **Solution**: Given: - $$a = 16$$ (shortest piece) - $$U_5 = 81$$ (longest piece) Visible text: - (shortest piece) - (longest piece) First step, determine the ratio: Component: MathContainer Children: ```math U_5 = a \cdot r^{5-1} ``` ```math 81 = 16 \cdot r^4 ``` ```math r^4 = \frac{81}{16} ``` ```math r = \sqrt[4]{\frac{81}{16}} = \frac{3}{2} ``` Then, find the length of the third piece ($$U_3$$): Visible text: Then, find the length of the third piece (): Component: MathContainer Children: ```math U_3 = a \cdot r^{3-1} ``` ```math U_3 = 16 \cdot \left(\frac{3}{2}\right)^2 ``` ```math U_3 = 16 \cdot \frac{9}{4} ``` ```math U_3 = 36 ``` Therefore, the length of the third piece is $$36 \text{ cm}$$. Visible text: Therefore, the length of the third piece is .