Lesson 2: Sequences and Series – AP and GP

👨🏽‍🏫 Introduction

Imagine you’re stacking blocks: 2 blocks in the first row, 4 in the second, 6 in the third… and so on. Or maybe you’re saving money each week, adding a fixed amount — or earning interest that grows your savings faster and faster. These patterns are called Sequences and Series. 🧱💰📈

Understanding sequences helps us recognize patterns, predict future values, and solve everyday problems like budgeting, investing, and analyzing data. In this lesson, you’ll explore:

• Arithmetic Progressions (AP)
• Geometric Progressions (GP)
• Formulas to find the n-th term and sum of terms
• How AP and GP compare in growth
• Common mistakes to avoid

🧠 Key Concepts

📌 1. What Is a Sequence?

A sequence is an ordered list of numbers that follow a pattern. Each number is called a term.

Examples:

• 2, 4, 6, 8, 10, … (Increasing by 2 — an AP)
• 3, 6, 12, 24, … (Multiplying by 2 — a GP)

🧾 2. Arithmetic Progression (AP)

An Arithmetic Progression (AP) is a sequence where each term is obtained by adding the same number, called the common difference (d), to the previous term.

General form:

• a = first term
• d = common difference (can be positive or negative)

Formula for the n-th term:

Formula for the sum of the first n terms:

💡 Why does it work? The n-th term formula adds the common difference (n – 1) times after the first term.

⚙️ 3. Geometric Progression (GP)

A Geometric Progression (GP) is a sequence where each term is obtained by multiplying by the same number, called the common ratio (r).

General form:

• a = first term
• r = common ratio (can be a fraction, whole number, or decimal)

Formula for the n-th term:

Sum of the first n terms (r ≠ 1):

💡 Why does it work? Each term is multiplied by the ratio one more time than the one before it — that’s why the exponent is (n – 1)!

📊 4. AP vs GP Growth – Visual Comparison

➡️ Conclusion: AP grows steadily. GP grows explosively!

🚨 Trap Alert – Common Mistakes

• ❌ Using n instead of n – 1 in formulas for the n-th term
• ❌ Forgetting to divide by 2 in the sum formula for AP
• ❌ Using a wrong sign for the common difference (d) when it’s negative
• ❌ Mixing up AP and GP formulas — always check if the pattern is additive or multiplicative!

🧮 Sample Problem Walkthroughs

1. Find the 10th term of the AP: 3, 7, 11, 15, …

   Solution: a = 3, d = 4

   

2. Find the sum of the first 6 terms of the AP: 5, 8, 11, 14, …

   Solution: a = 5, d = 3, n = 6

   

3. Find the 5th term of the GP: 2, 6, 18, …

   Solution: a = 2, r = 3

   

4. Find the sum of the first 4 terms of the GP: 1, 2, 4, 8

   Solution: a = 1, r = 2, n = 4

   

✍🏽 Practice Exercises

🅰️ A. Arithmetic Progression (AP):

1. Find the 12th term of the AP: 2, 5, 8, 11, …
2. Find the sum of the first 8 terms of the AP: 4, 10, 16, 22, …

🅱️ B. Geometric Progression (GP):

1. Find the 6th term of the GP: 3, 6, 12, …
2. Find the sum of the first 5 terms of the GP: 2, 4, 8, …

✅ Answers and Explanations

• A1: a = 2, d = 3

  

• A2: a = 4, d = 6, n = 8

  

• B1: a = 3, r = 2

  

• B2: a = 2, r = 2, n = 5

  

📌 Recap

• ✅ AP increases or decreases by adding a fixed number (common difference)
• ✅ GP grows by multiplying by a fixed number (common ratio)
• ✅ Use formulas to find the n-th term or sum efficiently
• ✅ Know the difference: AP = steady growth, GP = fast/exponential growth

💭 Reflection Prompt

Which type of growth do you think is more powerful in the long run — arithmetic or geometric? Can you think of a situation in life where this applies (e.g., savings vs. compound interest)?