Geometric Sequence

Multiplying Each Term by the Same Factor

In a geometric sequence, every term after the first is obtained by multiplying the previous term by a constant factor. This ratio is denoted by the letter $r$.

For nonzero terms, we can find that factor by division:

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:

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:

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

For a nonzero ratio and a positive integer term number, the general term is:

If $r=0$, use $U_1=a$ and $U_n=0$ for . This avoids evaluating for the first term.

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:

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:

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

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:

1. Population growth (as in the bacteria example)
2. Compound interest in economics
3. Radioactive decay in physics
4. 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:

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)

Each piece has a positive length, so the ratio is positive too. We therefore take the positive fourth root:

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

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