Geometric Sequence

Multiplying Each Term by the Same Factor

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:

\frac{U_2}{U_1} = \frac{U_3}{U_2} = \frac{U_4}{U_3} = ... = r

Each term is obtained by multiplying the previous term by the ratio $r$ . The quotient form above requires nonzero denominators, but the multiplication rule also works when a term is zero:

U_{n+1}=rU_n

Paper Folding Exploration

In a simple folding model, each fold doubles the number of layers. Fold a rectangular sheet of paper in half several times.

If the paper is folded $1 \text{ time}$ , it will be divided into $2 \text{ parts}$ of equal size. If folded again ( $2 \text{ times}$ ), it will form $4 \text{ parts}$ of equal size. 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 $2$ , so the ratio is $2$ .

General Formula for Geometric Sequences

The general formula for a geometric sequence is:

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

The quantities in this formula are:

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

The exponent is $n-1$ because moving from the first term to the $n$ th term takes exactly $n-1$ multiplications by $r$ .

Sequence Behavior for Different Ratios

For a positive first term, the ratio determines the overall behavior:

If $r>1$ , the positive terms grow.
If $0<r<1$ , the positive terms approach zero.
If $r=1$ , the sequence is constant.
If $r=0$ , every term after the first is zero.
If $r<0$ , the signs alternate while the magnitudes may grow, shrink, or stay constant.

This is why a geometric sequence does not always represent growth.

An Idealized Bacterial Growth Model

Suppose the number of bacteria in an ideal culture triples every $2 \text{ hours}$ . The factor $3$ describes how the total number of bacteria changes during each interval.

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

Initial count ( $N_0$ ) $=2$
Growth factor ( $r$ ) $=3$

In $20 \text{ hours}$ , the culture passes through $10 \text{ growth intervals}$ because $20 \div 2=10$ .

Starting at index $0$ keeps the initial count separate from the first completed growth interval. After $10$ intervals:

N_{10}=N_0\cdot r^{10}=2\cdot3^{10}=118{,}098

So, under this idealized model, the culture contains $118{,}098 \text{ bacterial cells}$ after $20 \text{ hours}$ .

Idealized Bacterial Growth

2 Bacteria

Ratio and General Term of a Geometric Sequence

Every step in a geometric sequence multiplies by the same factor. Dividing any term by the term before it recovers that factor, and one starting value together with one factor is enough to write the general term.

Ratio in Geometric Sequences

The factor $r$ in a geometric sequence is constant. Whenever the previous term is nonzero, the same factor 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}

Growth and Decay Models Using a Constant Ratio

Each process below repeats one multiplication at every step, so the same ratio drives the whole list:

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

A geometric sequence models repeated growth or decay only when the same ratio applies at every step.

Always check whether a constant-factor model is reasonable for the situation and for the time span being studied. Real populations eventually face limits that this simple model does not include.

Finding a Ratio and an Unknown Term

Both examples give part of a geometric sequence and ask for the missing value. Substitute the known terms into the general formula and solve for the unknown.

Finding the Ratio

The first term of a geometric sequence is $4$ and term $4$ is $108$ . Determine the ratio of this sequence.

Solution:

Given:

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

Using the general formula for geometric sequences:

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

The ratio of the geometric sequence is $3$ .

Length of Rope Sections

A rope is divided into $5 \text{ parts}$ whose positive lengths, ordered from shortest to longest, form a geometric sequence. The shortest piece is $16 \text{ cm}$ and the longest 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}

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

Multiplying Each Term by the Same Factor

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:

\frac{U_2}{U_1} = \frac{U_3}{U_2} = \frac{U_4}{U_3} = ... = r

Each term is obtained by multiplying the previous term by the ratio $r$ . The quotient form above requires nonzero denominators, but the multiplication rule also works when a term is zero:

U_{n+1}=rU_n

Paper Folding Exploration

In a simple folding model, each fold doubles the number of layers. Fold a rectangular sheet of paper in half several times.

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 $2$ , so the ratio is $2$ .

General Formula for Geometric Sequences

The general formula for a geometric sequence is:

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

The quantities in this formula are:

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

The exponent is $n-1$ because moving from the first term to the $n$ th term takes exactly $n-1$ multiplications by $r$ .

Sequence Behavior for Different Ratios

For a positive first term, the ratio determines the overall behavior:

If $r>1$ , the positive terms grow.
If $0<r<1$ , the positive terms approach zero.
If $r=1$ , the sequence is constant.
If $r=0$ , every term after the first is zero.
If $r<0$ , the signs alternate while the magnitudes may grow, shrink, or stay constant.

This is why a geometric sequence does not always represent growth.

An Idealized Bacterial Growth Model

Suppose the number of bacteria in an ideal culture triples every $2 \text{ hours}$ . The factor $3$ describes how the total number of bacteria changes during each interval.

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

Initial count ( $N_0$ ) $=2$
Growth factor ( $r$ ) $=3$

In $20 \text{ hours}$ , the culture passes through $10 \text{ growth intervals}$ because $20 \div 2=10$ .

Starting at index $0$ keeps the initial count separate from the first completed growth interval. After $10$ intervals:

N_{10}=N_0\cdot r^{10}=2\cdot3^{10}=118{,}098

So, under this idealized model, the culture contains $118{,}098 \text{ bacterial cells}$ after $20 \text{ hours}$ .

Idealized Bacterial Growth

2 Bacteria

Ratio and General Term of a Geometric Sequence

Ratio in Geometric Sequences

The factor $r$ in a geometric sequence is constant. Whenever the previous term is nonzero, the same factor 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}

Growth and Decay Models Using a Constant Ratio

Each process below repeats one multiplication at every step, so the same ratio drives the whole list:

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

A geometric sequence models repeated growth or decay only when the same ratio applies at every step.

Always check whether a constant-factor model is reasonable for the situation and for the time span being studied. Real populations eventually face limits that this simple model does not include.

Finding a Ratio and an Unknown Term

Both examples give part of a geometric sequence and ask for the missing value. Substitute the known terms into the general formula and solve for the unknown.

Finding the Ratio

The first term of a geometric sequence is $4$ and term $4$ is $108$ . Determine the ratio of this sequence.

Solution:

Given:

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

Using the general formula for geometric sequences:

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

The ratio of the geometric sequence is $3$ .

Length of Rope Sections

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}

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