Geometric sequences

Textbook reading

The problem that opens the lesson

A bacteria count is $600$ at hour $1$ and triples every $2$ hours. Write a sequence for counts at odd-numbered hours.

Solution

Begin by identifying the mathematical object and the information that fixes it. Use indexed terms to form $r^{difference$ in indices}, solve for r, and verify sign and magnitude against the term pattern. The relevant conditions are not optional bookkeeping: A zero term can make ratio-based recovery impossible or indicate a special sequence. Following that structure gives $a_n=600\cdot 3^{n-1}$ for hours $2n-1$ .

Why this works

Ratios may be positive, negative, fractional, or greater than one, creating growth, decay, or alternating signs. The calculation and the representation should agree, so a graph, diagram, table, or substitution check should support the same conclusion.

Textbook reading

What this lesson is really about

A geometric sequence has a constant ratio $r$ between consecutive nonzero terms.

From $a_1,$ reaching term $n$ requires $n-1$ multiplications, so $a_n=a_1 r^{n-1}$ . This is an exponential function sampled at integer inputs.

The point is not merely to reproduce a formula. A learner should be able to identify the quantities or geometric objects involved, explain why the relationship has its stated form, and recognize when the same idea appears in a graph, table, diagram, or model.

Textbook reading

Why the relationship works

Ratios may be positive, negative, fractional, or greater than one, creating growth, decay, or alternating signs.

Textbook reading

A reliable way to work

Use indexed terms to form $r^{difference$ in indices}, solve for r, and verify sign and magnitude against the term pattern.

A zero term can make ratio-based recovery impossible or indicate a special sequence.

After the symbolic work is complete, check the result. Depending on the lesson, this may mean substituting into an original equation, comparing coordinates, examining a graph, checking units, testing an interval, or confirming that every branch of a periodic solution has been included.

Textbook reading

What commonly goes wrong

A common error is dividing terms separated by several indices and treating the result as the one-step ratio.

The repair is to return to the definition and identify the first step where the invalid solution stops describing the original mathematical object. Later algebra cannot rescue a first step that changed the domain, orientation, branch, or meaning of the problem.

Textbook reading

Worked examples

Worked example 1

A bacteria count is $600$ at hour $1$ and triples every $2$ hours. Write a sequence for counts at odd-numbered hours.

Solution

Begin by identifying the mathematical object and the information that fixes it. Use indexed terms to form $r^{difference$ in indices}, solve for r, and verify sign and magnitude against the term pattern. The relevant conditions are not optional bookkeeping: A zero term can make ratio-based recovery impossible or indicate a special sequence. Following that structure gives $a_n=600\cdot 3^{n-1}$ for hours $2n-1$ .

Why this works

Ratios may be positive, negative, fractional, or greater than one, creating growth, decay, or alternating signs. The calculation and the representation should agree, so a graph, diagram, table, or substitution check should support the same conclusion.

Transfer example

Problem

Find $r$ from two consecutive terms.

Worked development

Use indexed terms to form $r^{difference$ in indices}, solve for r, and verify sign and magnitude against the term pattern. In this example, the first useful move is to make the defining structure visible rather than to search for a memorized answer. From $a_1,$ reaching term $n$ requires $n-1$ multiplications, so $a_n=a_1 r^{n-1}$ . This is an exponential function sampled at integer inputs. Then apply the conditions explicitly: A zero term can make ratio-based recovery impossible or indicate a special sequence. Finish by checking the result in a second representation and explaining what the result means.

Interpretation

Geometric sequences model compound growth, decay, rebounds, dilution, and repeated scaling.

Reasoning example

Problem

Find $r$ across a multi-step index gap.

Worked development

Use indexed terms to form $r^{difference$ in indices}, solve for r, and verify sign and magnitude against the term pattern. In this example, the first useful move is to make the defining structure visible rather than to search for a memorized answer. From $a_1,$ reaching term $n$ requires $n-1$ multiplications, so $a_n=a_1 r^{n-1}$ . This is an exponential function sampled at integer inputs. Then apply the conditions explicitly: A zero term can make ratio-based recovery impossible or indicate a special sequence. Finish by checking the result in a second representation and explaining what the result means.

Interpretation

Geometric sequences model compound growth, decay, rebounds, dilution, and repeated scaling.

Worked example 4: quick check

Find $a_8$ for $5,-10,20,$ ...

Solution

Begin by identifying the mathematical object and the information that fixes it. Use indexed terms to form $r^{difference$ in indices}, solve for r, and verify sign and magnitude against the term pattern. The relevant conditions are not optional bookkeeping: A zero term can make ratio-based recovery impossible or indicate a special sequence. Following that structure gives $-640$ .

Why this works

Ratios may be positive, negative, fractional, or greater than one, creating growth, decay, or alternating signs. The calculation and the representation should agree, so a graph, diagram, table, or substitution check should support the same conclusion.

Textbook reading

Application and interpretation

Geometric sequences model compound growth, decay, rebounds, dilution, and repeated scaling.

A contextual answer must include units, a meaningful domain, and the assumptions that make the model plausible. An exact mathematical relationship should not be diluted into a decimal unless a measurement or comparison requires it.

Practice