SL 1.3 : Geometric Sequences and Series

Content	Guidance, clarification and syllabus links
Geometric sequences and series. Use of the formulae for the nth term and the sum of the first n terms of the sequence. Use of sigma notation for sums of geometric sequences.	Spreadsheets, GDCs and graphing software may be used to generate and display sequences in several ways. If technology is used in examinations, students will be expected to identify the first term and the common ratio. Link to models/functions in topic 2 and regression in topic 4. Applications. Examples include the spread of disease, salary increase and decrease, and population growth.

📌 Introduction

Geometric sequences and series represent patterns where consecutive terms are related by multiplication by a constant factor called the common ratio. Unlike arithmetic sequences that grow by constant addition, geometric sequences exhibit exponential growth or decay patterns that model many real-world phenomena including population dynamics, radioactive decay, compound interest, and viral spread.

The power of geometric sequences lies in their multiplicative nature – once you know the first term and common ratio, you can determine any term in the sequence or calculate the sum of any number of terms. This exponential behavior makes geometric sequences essential for modeling situations with constant percentage growth or decay over equal time intervals.

📌 Definition Table

Term	Definition
Geometric Sequence	A sequence where the ratio between consecutive terms is constant. Example: 2, 6, 18, 54, 162, … (common ratio = 3)
Common Ratio (r)	The constant value by which each term is multiplied to get the next term: $r = \frac{u_{n+1}}{u_n}$ Can be positive, negative, fractional, or greater than 1
First Term (a or $u_1$)	The initial term of the sequence, often denoted as $u_1$ or $a$
nth Term ($u_n$)	General term of the sequence: $u_n = ar^{n-1}$ Also written as: $u_n = u_1 \times r^{n-1}$
Geometric Series	The sum of terms in a geometric sequence Example: 2 + 6 + 18 + 54 + 162 = 242
Sum of n terms ($S_n$)	$S_n = \frac{a(r^n – 1)}{r – 1}$ when $r \neq 1$ $S_n = na$ when $r = 1$
Sigma Notation ($\sum$)	Compact notation for expressing the sum of a series $\sum_{k=1}^{n} ar^{k-1} = a + ar + ar^2 + … + ar^{n-1}$

📌 Properties & Key Formulas

General Form: $u_1, u_1r, u_1r^2, u_1r^3, u_1r^4, \ldots$
nth Term Formula: $u_n = ar^{n-1}$ where $a$ is first term, $r$ is common ratio
Sum Formula (r ≠ 1): $S_n = \frac{a(r^n – 1)}{r – 1}$ or $S_n = \frac{a(1 – r^n)}{1 – r}$
Sum Formula (r = 1): $S_n = na$ (constant sequence)
Common Ratio: $r = \frac{u_2}{u_1} = \frac{u_3}{u_2} = \frac{u_4}{u_3} = \frac{u_{n+1}}{u_n}$
Sigma Notation: $\sum_{k=1}^{n} ar^{k-1} = \frac{a(r^n – 1)}{r – 1}$
Geometric Mean: If $a$, $b$, $c$ are consecutive terms, then $b^2 = ac$

Common Examples:

$3, 6, 12, 24, 48, \ldots$ (first term: 3, common ratio: 2)
$80, 40, 20, 10, 5, \ldots$ (first term: 80, common ratio: 0.5)
$1, -2, 4, -8, 16, \ldots$ (first term: 1, common ratio: -2)
$5, 5, 5, 5, 5, \ldots$ (first term: 5, common ratio: 1)

🧠 Examiner Tip: Always check that the ratio between consecutive terms is constant before applying geometric sequence formulas.

Remember: $r = \frac{\text{any term}}{\text{previous term}}$. If this ratio isn’t constant throughout the sequence, it’s not geometric!

📌 Common Mistakes & How to Avoid Them

⚠️ Common Mistake #1: Confusing geometric and arithmetic sequences

Wrong: Thinking 2, 4, 8, 16, … has common difference 2
Right: This sequence has common ratio 2 (each term is multiplied by 2)

How to avoid: Check if you add a constant (arithmetic) or multiply by a constant (geometric).

⚠️ Common Mistake #2: Using wrong exponent in nth term formula

Wrong: Using $u_n = ar^n$ instead of $u_n = ar^{n-1}$
Example: For sequence 3, 6, 12, 24, … finding $u_4$
Wrong: $u_4 = 3 \times 2^4 = 48$ ❌
Right: $u_4 = 3 \times 2^{4-1} = 3 \times 8 = 24$ ✅

How to avoid: Remember $u_1 = ar^{1-1} = ar^0 = a$, so the exponent is $(n-1)$.

⚠️ Common Mistake #3: Incorrect sum formula when r = 1

Wrong: Using $S_n = \frac{a(r^n – 1)}{r – 1}$ when $r = 1$
Right: When $r = 1$, use $S_n = na$

How to avoid: Always check the value of r first; if r = 1, the sequence is constant.

⚠️ Common Mistake #4: Sign errors with negative common ratios

Wrong: For sequence 2, -6, 18, -54, … claiming $r = 3$
Right: $r = \frac{-6}{2} = -3$

How to avoid: Carefully calculate $r = \frac{u_{n+1}}{u_n}$ including signs.

⚠️ Common Mistake #5: Mixing up sum formula versions

Wrong: Randomly choosing between $\frac{a(r^n – 1)}{r – 1}$ and $\frac{a(1 – r^n)}{1 – r}$
Right: Both are equivalent! Choose the one that’s easier for your calculation

How to avoid: Note that $\frac{a(r^n – 1)}{r – 1} = \frac{a(1 – r^n)}{1 – r}$ (multiply top and bottom by -1).

📌 Calculator Skills: Casio CG-50 & TI-84

📱 Using Casio CG-50 for Geometric Sequences

Method 1: Using TABLE function
1. Press [MENU] → Select “Graph”
2. Enter your sequence formula as Y1 = 3*(2^(X-1)) (for $u_n = 3 \times 2^{n-1}$)
3. Press [F6] (TABLE) to generate terms
4. Use [SET] to adjust starting value and step size

Method 2: Using List function
1. Press [MENU] → Select “Run-Matrix”
2. Press [OPTN] → [LIST] → [SEQ]
3. Enter: seq(3*(2^(X-1)), X, 1, 8, 1) for first 8 terms
4. Press [EXE] to calculate

Calculating sums:
1. Create list of terms using seq() function
2. Press [OPTN] → [LIST] → [SUM]
3. Apply to your list: sum(Ans)

Finding ratios:
• Create two lists: terms and their consecutive terms
• Use division to verify constant ratio

📱 Using TI-84 for Geometric Sequences

Setting up sequence mode:
1. Press [MODE], scroll to “SEQ” and press [ENTER]
2. Press [Y=] to access sequence editor

Entering sequences:
1. In Y= menu, enter: u(n) = 3*(2^(n-1))
2. Set nMin = 1 (starting value)
3. Press [2nd] → [WINDOW] to set sequence parameters

Generating terms from home screen:
1. Press [2nd] → [STAT] → [OPS] → [5:seq(]
2. Enter: seq(3*(2^(X-1)), X, 1, 8, 1)
3. Press [ENTER] for first 8 terms

Finding sums:
1. Press [2nd] → [STAT] → [MATH] → [5:sum(]
2. Enter: sum(seq(3*(2^(X-1)), X, 1, 8, 1))
3. Press [ENTER] to calculate sum

📱 Calculator Tips & Tricks

Verification method:
• Always check first few terms manually to verify your formula is correct
• Use calculator to find pattern, then verify with hand calculations

Working with fractional ratios:
• Use parentheses: (1/2)^(n-1) not 1/2^n-1
• Store fractions as decimals when necessary: 0.5^(n-1)

Large terms:
• For exponential growth, terms can become very large quickly
• Use scientific notation when appropriate
• Be aware of calculator limitations for very large numbers

📌 Mind Map

📌 Applications in Science and IB Math

Finance & Economics: Compound interest, investment growth, loan calculations, depreciation models
Biology & Medicine: Population growth, bacterial reproduction, virus spread models, drug elimination from body
Physics & Chemistry: Radioactive decay, nuclear reactions, chemical reaction rates, cooling/heating processes
Computer Science: Algorithm complexity, data compression, recursive algorithms, binary tree structures
Engineering: Signal processing, oscillations, mechanical vibrations, electrical circuit analysis
Environmental Science: Climate modeling, pollution spread, resource depletion, renewable energy calculations
Social Sciences: Social media spread, rumor propagation, market penetration models
Mathematics: Fractal geometry, infinite series, probability distributions, chaos theory

➗ IA Tips & Guidance: Use geometric sequences to model exponential growth and decay phenomena in your IA.

Excellent IA Topics:
• Compound interest vs simple interest analysis over long periods
• Population growth models using geometric sequences
• Radioactive decay and half-life calculations
• Viral spread modeling (especially relevant post-pandemic)
• Depreciation of assets using geometric decay
• Bacterial growth in laboratory conditions

IA Structure Tips:
• Collect real data and fit geometric models
• Compare theoretical predictions with actual data
• Discuss limitations of exponential models
• Use technology to generate and analyze large datasets
• Consider factors that might cause deviations from pure geometric growth

📌 Worked Examples (IB Style)

Q1. Find the 8th term of the geometric sequence: 5, 15, 45, 135, …

Solution:

Step 1: Identify the first term and common ratio
First term: $a = u_1 = 5$
Common ratio: $r = \frac{15}{5} = 3$

Step 2: Apply the nth term formula
$u_n = ar^{n-1}$

Step 3: Substitute values
$u_8 = 5 \times 3^{8-1} = 5 \times 3^7 = 5 \times 2187 = 10935$

✅ Final answer: $u_8 = 10935$

Q2. Find the sum of the first 6 terms of the sequence: 2, 6, 18, 54, …

Solution:

Step 1: Identify values
$a = 2$, $r = \frac{6}{2} = 3$, $n = 6$

Step 2: Choose appropriate sum formula
Since $r \neq 1$, use $S_n = \frac{a(r^n – 1)}{r – 1}$

Step 3: Substitute and calculate
$S_6 = \frac{2(3^6 – 1)}{3 – 1} = \frac{2(729 – 1)}{2} = \frac{2 \times 728}{2} = 728$

✅ Final answer: $S_6 = 728$

Alternative Method:
$S_6 = \frac{2(1 – 3^6)}{1 – 3} = \frac{2(1 – 729)}{-2} = \frac{2 \times (-728)}{-2} = 728$

Q3. Express the sum $4 + 12 + 36 + 108 + 324$ using sigma notation and find its value.

Solution:

Step 1: Find the pattern
$a = 4$, $r = \frac{12}{4} = 3$
General term: $u_n = 4 \times 3^{n-1}$

Step 2: Count the terms
Terms: 4, 12, 36, 108, 324 → 5 terms

Step 3: Write in sigma notation
$\sum_{k=1}^{5} 4 \times 3^{k-1}$

Step 4: Calculate the sum
$S_5 = \frac{4(3^5 – 1)}{3 – 1} = \frac{4(243 – 1)}{2} = \frac{4 \times 242}{2} = 484$

✅ Final answer: $\sum_{k=1}^{5} 4 \times 3^{k-1} = 484$

Q4. An investment of $1000 grows by 5% each year. Find the value after 10 years and the total growth over 10 years.

Solution:

Step 1: Identify the geometric sequence
This represents compound growth where:
$a = 1000$ (initial investment)
$r = 1.05$ (growth factor: 100% + 5% = 105%)
$n = 10$ (number of years)

Step 2: Find value after 10 years
$u_{11} = 1000 \times (1.05)^{10} = 1000 \times 1.629 = $1629$ (approximately)

Step 3: Find total growth
Total growth = Final value – Initial value
Total growth = $1629 – $1000 = $629

✅ Final answer: Value after 10 years = $1629, Total growth = $629

Real-world interpretation:
The investment grows exponentially, not linearly. Each year’s growth is 5% of the previous year’s total, creating compound growth.

Q5. Find the value of x if the terms 2x, 6x, and 18x form a geometric sequence.

Solution:

Step 1: Use the property of geometric sequences
For three consecutive terms in geometric sequence: $\frac{u_2}{u_1} = \frac{u_3}{u_2}$
Or equivalently: $u_2^2 = u_1 \times u_3$

Step 2: Set up the equation
Using the ratio property:
$\frac{6x}{2x} = \frac{18x}{6x}$

Step 3: Solve for x
$\frac{6x}{2x} = 3$ and $\frac{18x}{6x} = 3$
Both ratios equal 3, confirming this is geometric for any $x \neq 0$

Step 4: Verify the answer
The sequence 2x, 6x, 18x has common ratio 3 for any non-zero value of x
For example, when x = 1: sequence is 2, 6, 18 with ratio 3 ✓

✅ Final answer: Any value of x where $x \neq 0$

📝 Paper Tip: Always verify your answers by checking if the calculated terms actually form a geometric sequence with constant ratio.

Key verification steps:
• Check that consecutive ratios are equal
• Substitute back into original conditions
• Show clear working for finding first term and common ratio
• Use calculator to verify large calculations
• Consider special cases like r = 1 or negative ratios

📌 Multiple Choice Questions (with Detailed Solutions)

Q1. The 4th term of a geometric sequence is 24 and the 7th term is 192. What is the first term?

A) 3 B) 6 C) 8 D) 12

📖 Show Answer

Step-by-step solution:

1. Let first term = $a$, common ratio = $r$

2. $u_4 = ar^3 = 24$ … (1)

3. $u_7 = ar^6 = 192$ … (2)

4. Divide (2) by (1): $\frac{ar^6}{ar^3} = \frac{192}{24}$

5. $r^3 = 8$, so $r = 2$

6. Substitute into (1): $a \times 2^3 = 24$, so $a = 3$

✅ Answer: A) 3

Q2. A geometric sequence has first term 8 and common ratio 0.5. Which term is the first to be less than 0.1?

A) 6th term B) 7th term C) 8th term D) 9th term

📖 Show Answer

Step-by-step solution:

1. General term: $u_n = 8 \times (0.5)^{n-1}$

2. For term less than 0.1: $8 \times (0.5)^{n-1} < 0.1$

3. $(0.5)^{n-1} < 0.0125$

4. Testing values: $u_7 = 8 \times (0.5)^6 = 8 \times 0.015625 = 0.125$

5. $u_8 = 8 \times (0.5)^7 = 8 \times 0.0078125 = 0.0625$

✅ Answer: C) 8th term

Q3. Which of the following represents $\sum_{k=1}^{6} 2 \times 3^{k-1}$?

A) 2 + 6 + 18 + 54 + 162 + 486 B) 2 + 6 + 18 + 54 + 162
C) 6 + 18 + 54 + 162 + 486 + 1458 D) 3 + 6 + 12 + 24 + 48 + 96

📖 Show Answer

Step-by-step solution:

1. Find each term: when $k=1$, $2 \times 3^{1-1} = 2 \times 3^0 = 2$

2. when $k=2$, $2 \times 3^{2-1} = 2 \times 3^1 = 6$

3. when $k=6$, $2 \times 3^{6-1} = 2 \times 3^5 = 486$

✅ Answer: A) 2 + 6 + 18 + 54 + 162 + 486

📌 Short Answer Questions (with Detailed Solutions)

Q1. The first three terms of a geometric sequence are $x + 1$, $2x + 2$, and $6x – 3$. Find the value of $x$.

📖 Show Answer

Complete solution:

Step 1: Use the property that common ratio must be constant

For geometric sequence: $\frac{2x + 2}{x + 1} = \frac{6x – 3}{2x + 2}$

Step 2: Cross-multiply

$(2x + 2)^2 = (x + 1)(6x – 3)$

$4x^2 + 8x + 4 = 6x^2 – 3x + 6x – 3$

$4x^2 + 8x + 4 = 6x^2 + 3x – 3$

Step 3: Solve quadratic

$0 = 2x^2 – 5x – 7 = (2x – 7)(x + 1)$

So $x = \frac{7}{2}$ or $x = -1$

If $x = -1$, first term = 0 (degenerate case)

✅ Answer: $x = \frac{7}{2}$

Q2. A geometric sequence has $u_2 = 12$ and $u_5 = 96$. Find the first term and common ratio.

📖 Show Answer

Complete solution:

Step 1: Set up equations

$u_2 = ar = 12$ … (1)

$u_5 = ar^4 = 96$ … (2)

Step 2: Find common ratio

Divide (2) by (1): $\frac{ar^4}{ar} = \frac{96}{12}$

$r^3 = 8$, so $r = 2$

Step 3: Find first term

Substitute into (1): $a \times 2 = 12$, so $a = 6$

✅ Answer: $a = 6$, $r = 2$

📌 Extended Response Questions (with Full Solutions)

Q1. A viral video is shared according to a geometric pattern. On day 1, the video has 100 views. Each day, the number of new views is 2.5 times the number of new views from the previous day.

(a) Show that the number of new views on day $n$ is given by $u_n = 100 \times (2.5)^{n-1}$. [2 marks]

(b) Find the total number of views after 7 days. [4 marks]

📖 Show Answer

Complete solution:

(a) Showing the formula

Day 1: 100 new views

Day 2: $100 \times 2.5 = 250$ new views

Day 3: $250 \times 2.5 = 100 \times (2.5)^2 = 625$ new views

This forms a geometric sequence with $a = 100$, $r = 2.5$

General term: $u_n = ar^{n-1} = 100 \times (2.5)^{n-1}$ ✓

(b) Total views after 7 days

Using sum formula: $S_n = \frac{a(r^n – 1)}{r – 1}$

$S_7 = \frac{100((2.5)^7 – 1)}{2.5 – 1} = \frac{100(610.35 – 1)}{1.5} = 40,623$ views

We need: $100 \times (2.5)^{n-1} > 50,000$

$(2.5)^{n-1} > 500$

Taking logarithms: $(n-1)\ln(2.5) > \ln(500)$

$n-1 > \frac{6.21}{0.916} ≈ 6.78$

So $n > 7.78$, meaning day 8

✅ Final Answers:
(a) Shown: $u_n = 100 \times (2.5)^{n-1}$
(b) 40,623 total views after 7 days
(c) Day 8

Term	Definition
Geometric Sequence	A sequence where the ratio between consecutive terms is constant. Example: 2, 6, 18, 54, 162, … (common ratio = 3)
Common Ratio (r)	The constant value by which each term is multiplied to get the next term: \(r = \frac{u_{n+1}}{u_n}\) Can be positive, negative, fractional, or greater than 1
First Term (a or \(u_1\))	The initial term of the sequence, often denoted as \(u_1\) or \(a\)
nth Term (\(u_n\))	General term of the sequence: \(u_n = ar^{n-1}\) Also written as: \(u_n = u_1 \times r^{n-1}\)
Geometric Series	The sum of terms in a geometric sequence Example: 2 + 6 + 18 + 54 + 162 = 242
Sum of n terms (\(S_n\))	\(S_n = \frac{a(r^n – 1)}{r – 1}\) when \(r \neq 1\) \(S_n = na\) when \(r = 1\)
Sigma Notation (\(\sum\))	Compact notation for expressing the sum of a series \(\sum_{k=1}^{n} ar^{k-1} = a + ar + ar^2 + … + ar^{n-1}\)