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Sequences and Series

Geometric Sequence

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 $r$ .

If we have a geometric sequence $U_1, U_2, U_3, ..., U_n$ , then:

Geometric SequenceLearn geometric sequences with constant ratios. Learn formulas, solve real-world problems like bacterial growth, and discover practical applications.

On this page

Understanding Geometric Sequences
Paper Folding Exploration
General Formula for Geometric Sequences
Bacterial Division
Properties of Geometric Sequences
- Ratio in Geometric Sequences
- Finding the nth Term
Applications of Geometric Sequences
Example Problems
- Finding the Ratio
- Length of Rope Sections

Comments

\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 $r$ .

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:

Number of folds	Number of equal parts
$1 \text{ fold}$	$2 \text{ parts}$
$2 \text{ folds}$	$4 \text{ parts}$
$3 \text{ folds}$	$8 \text{ parts}$
$4 \text{ folds}$	$16 \text{ parts}$

Notice that the number of parts formed creates a sequence: $2, 4, 8, 16, \ldots$

In this sequence, each term is obtained by multiplying the previous term by $2$ . In other words, the ratio is $2$ .

General Formula for Geometric Sequences

The general formula for a geometric sequence is:

U_n = a \cdot r^{n-1}

Where:

$U_n$ = general term
$a$ = first term
$r$ = ratio
$n$ = term number

Bacterial Division

Bacteria reproduce by dividing themselves. Within two hours, one bacterial cell divides into $3 \text{ parts}$ .

If the initial number of bacteria is $2 \text{ cells}$ , then:

First term ( $U_1$ ) $4$
Ratio ( $r$ ) $= 3$

In $20 \text{ hours}$ , division occurs $10\text{ times}$ ( $20 \div 2 = 10$ ).

To determine the number of bacteria after $20 \text{ hours}$ ( $10$ th term), we use the formula:

U_{10} = 2 \cdot 3^{10-1} = 2 \cdot 3^9 = 2 \cdot 19{,}683 = 39{,}366

So, after $20 \text{ hours}$ , there are $39{,}366 \text{ bacterial cells}$ .

Bacterial Division

2 Bacteria

Properties of Geometric Sequences

Ratio in Geometric Sequences

The ratio ( $3 \text{ parts}$ ) in a geometric sequence is always constant and can be calculated by dividing the next term by the previous term:

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:

U_n = a \cdot r^{n-1}

Applications of Geometric Sequences

Geometric sequences are applied in various fields, such as:

Population growth (as in the bacteria example)
Compound interest in economics
Radioactive decay in physics
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.

Solution:

Given:

$a = 4$ (first term)
$U_4 = 108$ (fourth term)

Using the general formula for geometric sequences:

U_4 = a \cdot r^{4-1}

Therefore, the ratio of the geometric sequence is $3$ .

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.

Solution:

Given:

$a = 16$ (shortest piece)
$U_5 = 81$ (longest piece)

First step, determine the ratio:

U_5 = a \cdot r^{5-1}

Then, find the length of the third piece ( $U_3$ ):

U_3 = a \cdot r^{3-1}

Therefore, the length of the third piece is $36 \text{ cm}$ .

Arithmetic Sequence

Difference between Arithmetic and Geometric Sequence