Chapter 5: Progressions

Overview

Progressions (or sequences) are fundamental mathematical concepts that describe ordered lists of numbers following specific patterns. This chapter explores two main types of progressions: arithmetic progressions (AP) and geometric progressions (GP). Understanding these sequences is crucial for solving problems involving patterns, series, and real-world applications such as financial planning, population growth, and physics calculations.

Learning Objectives

After completing this chapter, you will be able to:

• Identify and classify arithmetic and geometric progressions
• Apply formulas for nth term and sum of terms
• Solve problems involving infinite geometric series
• Apply progression concepts in real-world scenarios
• Master SPM examination techniques for progression problems

Key Concepts

5.1 Arithmetic Progressions (AP)

Definition

An arithmetic progression is a sequence of numbers where the difference between consecutive terms is constant. This constant difference is called the common difference (d).

Example: 2, 5, 8, 11, 14, ... (common difference d = 3)

General Form

An AP can be written as: a, a + d, a + 2d, a + 3d, ..., a + (n-1)d

Where:

• a = first term
• d = common difference
• n = term number

Key Formulas

1. nth Term: $T_n = a + (n-1)d$
2. Sum of First n Terms:
   • $S_n = \frac{n}{2} [2a + (n-1)d]$
   • $S_n = \frac{n}{2} [a + l]$ (where l is the last term)

5.2 Geometric Progressions (GP)

Definition

A geometric progression is a sequence of numbers where the ratio between consecutive terms is constant. This constant ratio is called the common ratio (r).

Example: 3, 6, 12, 24, 48, ... (common ratio r = 2)

General Form

A GP can be written as: a, ar, a$r^2$, a$r^3$, ..., ar^(n-1)

Where:

• a = first term
• r = common ratio
• n = term number

Key Formulas

1. nth Term: $T_n = ar^{n-1}$
2. Sum of First n Terms:
   • $S_n = \frac{a(r^n - 1)}{r - 1}$ for $|r| > 1$
   • $S_n = \frac{a(1 - r^n)}{1 - r}$ for $|r| < 1$
3. Sum to Infinity: $S_\infty = \frac{a}{1 - r}$ (only valid if $|r| < 1$)

Important Formulas and Methods

Arithmetic Progression Formulas

Finding Common Difference: $d = T_n - T_{n-1}$

Finding Number of Terms: $n = \frac{l - a}{d} + 1$ (where l is last term)

Properties of AP:

• The middle term of three consecutive terms is the arithmetic mean
• The sum of terms equidistant from the beginning and end is constant

Geometric Progression Formulas

Finding Common Ratio: $r = \frac{T_n}{T_{n-1}}$

Infinite Series Convergence: A GP converges to a finite sum only if $|r| < 1$

Properties of GP:

• The middle term of three consecutive terms is the geometric mean
• The product of terms equidistant from the beginning and end is constant

Problem-Solving Strategies

AP Problems:

• Identify the common difference first
• Use nth term formula for specific term problems
• Use sum formulas for total sum problems
• Be careful with term numbering (first term is n=1)

GP Problems:

• Check the common ratio carefully
• Verify convergence conditions for infinite series
• Use logarithms for problems involving exponents

Solved Examples

Example 1: Arithmetic Progression

Given the AP: 5, 9, 13, 17, ... Find: a) The 15th term b) The sum of the first 15 terms c) Which term is 101?

Solutions:

a) First term a = 5, common difference d = 9 - 5 = 4 T_15 = a + (15-1)d = 5 + 14×4 = 5 + 56 = 61

b) S_15 = 15/2 [2×5 + (15-1)×4] = 15/2 [10 + 56] = 15/2 × 66 = 15 × 33 = 495

c) T_n = 101 ⇒ 5 + (n-1)×4 = 101 (n-1)×4 = 96 ⇒ n-1 = 24 ⇒ n = 25

Example 2: Geometric Progression

Given the GP: 2, 6, 18, 54, ... Find: a) The 8th term b) The sum of the first 8 terms c) The sum to infinity

Solutions:

a) First term a = 2, common ratio r = 6/2 = 3 T_8 = ar^(8-1) = 2 × 3^7 = 2 × 2187 = 4374

b) S_8 = a(r^n - 1)/(r - 1) = 2(3^8 - 1)/(3 - 1) = 2(6561 - 1)/2 = 6560

c) For sum to infinity, |r| < 1 must hold. Here r = 3 > 1, so the series diverges (no finite sum to infinity).

Example 3: Mixed Problem

The 4th term of an AP is 20 and the 9th term is 41. Find: a) The first term and common difference b) The sum of the first 20 terms

Solutions:

a) Let first term = a, common difference = d T_4 = a + 3d = 20 T_9 = a + 8d = 41

Subtract first equation from second: (a + 8d) - (a + 3d) = 41 - 20 5d = 21 ⇒ d = 21/5 = 4.2

Then a + 3(4.2) = 20 ⇒ a + 12.6 = 20 ⇒ a = 7.4

b) S_20 = 20/2 [2×7.4 + (20-1)×4.2] = 10 [14.8 + 79.8] = 10 × 94.6 = 946

Example 4: Geometric Series with Infinite Sum

Find the sum to infinity of the GP: 8 + 4 + 2 + 1 + ...

Solution: First term a = 8, common ratio r = 4/8 = 0.5 Since |r| = 0.5 < 1, the series converges.

S_∞ = a/(1 - r) = 8/(1 - 0.5) = 8/0.5 = 16

Example 5: Word Problem Applications

A person saves RM100 in the first month, RM110 in the second month, RM120 in the third month, and so on. How much will they have saved after 2 years? What is the total amount saved?

Solution: This is an AP with a = 100, d = 10, n = 24 months After 2 years = 24 months

S_24 = 24/2 [2×100 + (24-1)×10] = 12 [200 + 230] = 12 × 430 = RM5160

Mathematical Derivations

Derivation of AP Sum Formula

Let S_n = a + (a + d) + (a + 2d) + ... + [a + (n-1)d]

Write it in reverse: S_n = [a + (n-1)d] + [a + (n-2)d] + ... + a

Add the two equations: 2S_n = [2a + (n-1)d] + [2a + (n-1)d] + ... + [2a + (n-1)d] 2S_n = n[2a + (n-1)d] S_n = n/2 [2a + (n-1)d]

Derivation of GP Sum Formula

Let S_n = a + ar + a$r^2$ + ... + ar^(n-1)

Multiply by r: rS_n = ar + a$r^2$ + ... + ar^n

Subtract: S_n - rS_n = a - ar^n S_n(1 - r) = a(1 - r^n) S_n = a(1 - r^n)/(1 - r)

For |r| < 1, as n → ∞, r^n → 0 Therefore, S_∞ = a/(1 - r)

Derivation of nth Term Formulas

AP nth Term: T_n = a + (n-1)d This follows directly from the pattern: first term + (n-1) steps of size d

GP nth Term: T_n = ar^(n-1) This follows from multiplying the first term by the common ratio (n-1) times

Real-World Applications

1. Financial Mathematics - Compound Interest

Compound interest creates a geometric progression:

• Amount after n years: A = P(1 + r/100)^n
• Where P = principal, r = annual interest rate

Example: RM1000 invested at 5% annual compound interest Amount after 5 years: A = 1000(1.05)^5 ≈ RM1276.28

2. Physics - Exponential Decay

Radioactive decay follows exponential patterns:

• Remaining amount: N = $N_0$ e^(-λt)
• Half-life calculations use geometric progression

Example: Substance with half-life 5 years. Initial amount 100g After 15 years: N = 100 × (1/2)^(15/5) = 100 × (1/2)^3 = 12.5g

3. Population Growth

Population growth can be modeled as geometric progression:

• Population after n years: P = $P_0$(1 + r)^n
• Where r = growth rate

Example: City population 1,000,000 growing at 2% annually Population after 10 years: P = 1,000,000(1.02)^10 ≈ 1,218,994

4. Depreciation

Asset value depreciation follows geometric progression:

• Value after n years: V = $V_0$(1 - d)^n
• Where d = depreciation rate

Example: Car worth RM50,000 depreciating at 15% annually Value after 5 years: V = 50,000(0.85)^5 ≈ RM22,091.25

Complex Problem-Solving Techniques

Problem: Three numbers in AP have sum 30 and product 960. Find the numbers.

Solution: Let the three numbers be a-d, a, a+d (since they're in AP) Sum: (a-d) + a + (a+d) = 3a = 30 ⇒ a = 10 Product: (a-d) × a × (a+d) = a($a^2$ - $d^2$) = 960 10(100 - $d^2$) = 960 ⇒ 100 - $d^2$ = 96 ⇒ $d^2$ = 4 ⇒ d = ±2

Numbers: 8, 10, 12 or 12, 10, 8

Problem: Find the sum of the first n terms of the series: 1 + 11 + 111 + 1111 + ...

Solution: Each term can be written as: 1 = (10 - 1)/9 11 = (100 - 1)/9 111 = (1000 - 1)/9 1111 = (10000 - 1)/9

So the sum S_n = Σ[(10^k - 1)/9] for k=1 to n = (1/9)[Σ10^k - Σ1] = (1/9)[(10(10^n - 1)/9) - n]

Problem: A bouncing ball rebounds to 3/4 of its previous height. If dropped from 10m, find:

a) The total distance traveled until it stops bouncing b) The height after 10 bounces

Solution: a) The ball travels: 10 + 2(7.5 + 5.625 + 3.759 + ...) First drop: 10m Then up and down: 2 × [7.5 + 5.625 + 3.759 + ...] (GP with a = 7.5, r = 0.75)

Total distance = 10 + 2 × [7.5/(1 - 0.75)] = 10 + 2 × 30 = 70m

b) Height after 10 bounces: h = 10 × (0.75)^10 ≈ 0.563m

Summary Points

• Arithmetic progressions have constant differences between terms
• Geometric progressions have constant ratios between terms
• Master the nth term and sum formulas for both types
• Infinite geometric series converge only if |r| < 1
• Applications include finance, physics, population studies, and depreciation
• Careful identification of the first term and common difference/ratio is crucial

Common Mistakes to Avoid

1. Term numbering errors - Remember first term is n=1, not n=0
2. Common ratio calculation - Be careful with division and signs
3. Infinite series conditions - Always check |r| < 1 before using sum to infinity
4. Mixed sequences - Don't confuse AP and GP formulas
5. Word problem setup - Correctly identify the type of progression from the description

SPM Exam Tips

Exam Strategies

1. Memorize formulas correctly - AP and GP formulas are different
2. Identify the type quickly - Look for constant difference or ratio
3. Check convergence conditions - Especially for infinite series
4. Handle word problems systematically - Extract a, d/r, n from the problem
5. Work step by step - Show clear working for partial marks

Key Exam Topics

• AP nth term calculation (20% of questions)
• AP sum calculation (25% of questions)
• GP nth term calculation (15% of questions)
• GP sum calculation (20% of questions)
• Infinite series (10% of questions)
• Word problems (10% of questions)

Time Management Tips

• Basic AP problems: 2-3 minutes
• Basic GP problems: 3-4 minutes
• Sum calculations: 4-5 minutes
• Infinite series: 2-3 minutes
• Word problems: 6-8 minutes

Practice Problems

Level 1: Arithmetic Progressions

1. For the AP: 3, 7, 11, 15, ... a) Find the 12th term b) Find the sum of first 12 terms c) Which term is 119?

2. The 3rd term of an AP is 14 and the 10th term is 41. Find: a) First term and common difference b) Sum of first 20 terms

Level 2: Geometric Progressions

1. For the GP: 4, 12, 36, 108, ... a) Find the 6th term b) Find the sum of first 6 terms c) Does the sum to infinity exist? If yes, find it.

2. Find the sum to infinity of: a) 6 + 3 + 1.5 + 0.75 + ... b) 10 - 5 + 2.5 - 1.25 + ...

Level 3: Mixed Problems

1. Three consecutive terms in GP are x, 2x+3, 4x+1. Find x and the terms.

2. The sum of three numbers in AP is 33 and their product is 1287. Find the numbers.

3. A GP has first term 2 and common ratio 1.5. Find: a) The 8th term b) The sum of first 8 terms c) Which term first exceeds 1000?

Level 4: Applications

1. Investment: RM5000 invested at 6% annual compound interest. Find amount after 8 years.

2. Population: Town population 50,000 decreases by 5% annually. Find population after 10 years.

3. Physics: Ball dropped from 20m, rebounds to 80% of previous height. Find: a) Total distance until it stops bouncing b) Height after 15 bounces

4. Finance: Savings plan - RM100 first month, increases by RM10 each month. Find total after 3 years.

Did You Know? 📚

The concept of arithmetic sequences dates back to ancient Babylonian mathematics around 2000 BCE. The famous mathematician Carl Friedrich Gauss (1777-1855) famously solved the sum of integers from 1 to 100 as a child by recognizing it was an arithmetic series, a story that illustrates the power of understanding mathematical patterns.

Quick Reference Guide

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Progressions are essential for understanding patterns in mathematics and their applications in real-world scenarios. Mastery of these concepts will serve as a foundation for more advanced topics in calculus and series analysis.