Model geometric sequences and series

International Baccalaureate (IB) IB AA SL: Mathematics: Analysis and Approaches SL

Number and Algebra

Recognising multiplicative change, finding terms and totals, and choosing a suitable model

A geometric sequence describes a pattern in which each term is found by multiplying the previous term by the same number. This makes it useful for situations such as repeated percentage change or repeated scaling. A sequence lists terms; a series adds terms. In this lesson, the first term is written as

u_1

, the common ratio as

r

, and the number of terms as

n

. A ratio may be negative or fractional, so consider whether the resulting pattern makes sense in its context.

What you will learn

Recognise a geometric sequence and identify its first term and common ratio.
Use a term rule and finite-sum rule to answer questions about geometric patterns.
Model repeated percentage change and interpret the model in context.
Use a calculator or graphing technology to check a result while explaining the mathematics.

1. Prior knowledge and recognising the pattern

You should be comfortable simplifying fractions, substituting values into formulas, and solving a simple equation. To test whether a sequence is geometric, divide each term by the one before it. If the resulting ratios are the same, that fixed value is the common ratio

r

. For example, the terms CAD 5, 15, 45 have ratio

3

each time.

The first term is

u_1

. Multiplying it by

r

gives the second term; multiplying again gives the third. Repeating this pattern gives the term rule. The index

n

tells you the position of a term, so it is a positive integer. A sequence with a negative ratio alternates signs; one with a ratio between

0

and

1

in magnitude shrinks in magnitude.

r=\frac{u_{n+1}}{u_n}

Check consecutive ratios, not consecutive differences.
State what each parameter means before using a model.
The sequence is discrete: its terms occur at integer positions.

2. Term rules and representations

Once the first term and common ratio are known, the term at position

n

can be found directly. This saves listing every earlier term. For example, when

u_1=4

and

r=2

, the sixth term is

4\times 2^5

, because there are five multiplications after the first term.

The same model has numerical, symbolic, graphical, and contextual forms. A list of terms is numerical; the term rule is symbolic; plotting the points

(n,u_n)

is graphical; and a statement such as “the amount doubles each week” gives a context. The graph consists of separate points because

n

counts whole positions, not every real number.

For repeated percentage change, convert the percentage to a multiplier. An increase of p% uses multiplier

1+\frac{p}{100}

; a decrease uses

1-\frac{p}{100}

. The starting amount is

u_1

if the first time point in the model is labelled

1

. If a context labels the initial amount as time zero, write the model with that starting value at time zero instead.

u_n=u_1r^{n-1}

The exponent is one less than the term position when indexing starts at 1.
A percentage change is applied multiplicatively at each equal time interval.
Check that the index and time labels match the context.

3. Finite sums, infinite sums, and checking a model

A geometric series is the sum of terms from a geometric sequence. For a finite series with first term

u_1

and

n

terms, the sum can be calculated without adding each term one by one. The formula applies when

r\ne1

. If

r=1

, every term is

u_1

, so the sum is

nu_1

.

An infinite geometric series has a finite sum only when the common ratio has magnitude less than

1

. In that case, later terms get closer to zero, and adding indefinitely many terms approaches a fixed total. If

|r|\ge1

, do not use the infinite-sum formula: the terms do not approach zero.

A graphing calculator can check a finite model by generating a sequence or plotting points for integer values of

n

. It can also calculate a finite sum. Enter the model and bounds carefully, and keep the analytical reasoning: identify

u_1

,

r

, and

n

, choose the relevant formula, and interpret the result with units and suitable rounding. A calculator display is a check, not an explanation.

S_n=\frac{u_1(1-r^n)}{1-r}\quad(r\ne1),\qquad S_\infty=\frac{u_1}{1-r}\quad(|r|<1)

Use a finite-sum model when the question asks for a total over a fixed number of terms.
Use an infinite sum only if the common ratio satisfies $|r|<1$ .
Round only at the end, and include units when the context has units.

Worked example

Finding a distant term

A geometric sequence begins

7, 21, 63,\ldots

. Find the eighth term.

Identify the parameters
Divide a term by the preceding term to find the repeated multiplier. The first term is $7$ and the common ratio is $3$ .
$u_1=7,\qquad r=3$
Substitute the position
Use the term rule with $n=8$ . The exponent is $8-1$ because reaching the eighth term requires seven multiplications after the first term.
$u_8=7(3)^{8-1}$
Evaluate
Calculate the power and multiply by the first term.
$u_8=7(2187)=15309$

Answer: The eighth term is

15309

.

Check: The first three terms are obtained by multiplying by

3

, which agrees with the given sequence. The eighth term is positive, as expected for this pattern.

Worked example

Repeated percentage decrease

A machine is worth CAD 24,000 initially. Its value decreases by 12% at the end of each year. Model its value after five years and give the value to the nearest dollar.

Find the annual multiplier
A 12% decrease leaves 88% of the value from the previous year. Thus each year's value is found by multiplying by $0.88$ .
$r=1-\frac{12}{100}=0.88$
Write the model
Let $V_t$ be the value in dollars after $t$ years, with the initial value at $t=0$ . There are five multiplications after five years.
$V_t=24000(0.88)^t$
Evaluate and round
Substitute $t=5$ . Keep calculator precision until the final step, then round to the nearest dollar.
$V_5=24000(0.88)^5\approx12665.57$

Answer: After five years, the model gives a value of CAD 12,666 to the nearest dollar.

Check: The value is below CAD 24,000 because the multiplier is less than

1

. A graphing calculator can plot

V_t

at integer years to confirm the decreasing pattern.

Worked example

Adding a finite geometric series

A display has 6 rows of lights. The first row has 5 lights, and each following row has twice as many as the previous row. Find the total number of lights.

Identify the sequence
The row counts form a geometric sequence with first term $5$ , common ratio $2$ , and six terms.
$u_1=5,\qquad r=2,\qquad n=6$
Choose the finite-sum rule
The question asks for the total across a fixed number of rows, so use the formula for the sum of six terms.
$S_6=\frac{5(1-2^6)}{1-2}$
Calculate the total
Evaluate the expression. The denominator is negative, and the numerator is also negative, so the total is positive.
$S_6=315$

Answer: There are 315 lights in total.

Check: The row counts are 5, 10, 20, 40, 80, and 160 lights. Their sum is

315

, matching the formula.

Common mistakes and how to avoid them

Using a common difference instead of a common ratio.

Correction: For a geometric sequence, divide consecutive terms. Equal differences describe a different type of sequence.

Using an exponent of

n

in the term rule when the first term is indexed by 1.

Correction: Use exponent

n-1

: the first term has had zero multiplications by the ratio.

Using the finite-sum formula for an infinite series when

|r|\ge1

.

Correction: Check the condition first. The infinite series has a finite sum only when

|r|<1

.

Treating a percentage decrease of 12% as a multiplier of 0.12.

Correction: A 12% decrease leaves 88% of the previous amount, so the multiplier is 0.88.

Lesson summary

A geometric sequence has a constant ratio between consecutive terms.
Use the first term and common ratio to find any term directly.
A finite geometric series adds a fixed number of terms; an infinite geometric series has a finite sum only when $|r|<1$ .
For a contextual model, define the variables, match the time indexing, and state units and rounding.

Check your understanding

Question 1

What is the common ratio of

12, 6, 3, \ldots

?

$2$
$\frac{1}{2}$
$-6$
$-\frac{1}{2}$

Show answer and explanation

\frac{1}{2}

Divide a term by the preceding term:

6\div12=\frac{1}{2}

, and

3\div6=\frac{1}{2}

.

Question 2

A sequence has

u_1=3

and

r=2

. What is

u_5

?

$24$
$48$
$96$
$15$

Show answer and explanation

48

Use

u_5=3(2)^{5-1}=48

.

Question 3

Which common ratio allows an infinite geometric series with a finite sum?

$-1.2$
$1$
$0.6$
$-1$

Show answer and explanation

0.6

The required condition is

|r|<1

. Only

0.6

among the options satisfies it.

Key terms

Geometric sequence: A sequence in which each term after the first is found by multiplying the preceding term by a fixed common ratio.
Common ratio: The constant multiplier between consecutive terms of a geometric sequence.
Geometric series: The sum of one or more terms of a geometric sequence.
Finite series: A series that contains a fixed, limited number of terms.
Infinite series: A series formed by adding terms without a final term; a geometric infinite series has a finite sum only when the common ratio has magnitude less than 1.

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Published by DoAssignment. This reviewed lesson follows International Baccalaureate (IB) IB AA SL: Mathematics: Analysis and Approaches SL, study topic SL 1.3. It is a study resource, not an official curriculum publication.

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