General Mathematics Complete Reviewer

Week 6 · General Mathematics

Percentages in Business & Finance

Inflation · Mark-Up & Discount · Value-Added Tax (VAT) · Profit & Loss

Learning Objectives

Define inflation, mark-up, discount, VAT, profit, and loss and identify their applications in everyday situations.
Solve problems involving percentage increases/decreases, mark-ups, discounts, VAT, profit, and loss.
Analyze how inflation, mark-ups, and deductions affect consumer choices and purchasing power.
Prepare a Business Scenario Analysis showing the impact of these concepts on financial decisions.

Topic 1 · Inflation

What is Inflation?

Inflation is the general increase in prices of goods and services over time, which causes the value of money to go down. It affects purchasing power — it is important to track price changes for better financial decisions.

Percentage Increase / Decrease Formula

% Change = [(New − Old) / Old] × 100

📌 Example 1

Rice was ₱40/kilo last year. This year it is ₱48. What is the percentage increase?

% Increase = [(48 − 40) / 40] × 100 = [8/40] × 100 = 20%

📌 Example 2

Father earned ₱15,000 last year; now ₱15,750. Inflation is 6%. Did his salary keep up?

Salary increase = [(15,750 − 15,000) / 15,000] × 100 = 5%
Inflation = 6% → His salary did NOT keep up with inflation.

Topic 2 · Mark-Up & Discount

Key Terms & Formulas

Term	Definition	Formula
Mark-Up	Amount added to cost price to earn profit	`Mark-Up = SP − CP`
Discount	Amount reduced from the original price during a sale	`Discount = Original − Net Price`
Selling Price (SP)	Final price charged to the customer	`SP = CP + Mark-Up`
Net Price	Actual amount paid after discount	`Net Price = Original − Discount`
Mark-Up %	Mark-up as a percentage of cost price	`Mark-Up % = (Mark-Up / CP) × 100`
Discount %	Discount as a percentage of original price	`Discount % = (Discount / Original) × 100`

📌 Example 3 – Mark-Up

Store buys a bag for ₱500 and adds 30% mark-up. What is the selling price?

Mark-Up = 30% × 500 = ₱150 → SP = 500 + 150 = ₱650

📌 Example 4 – Discount

₱1,000 jacket is on 20% discount. What is the net price?

Discount = 20% × 1,000 = ₱200 → Net Price = 1,000 − 200 = ₱800

📌 Example 5 – Mark-Up %

Item bought for ₱400 sold for ₱520. What is the mark-up percentage?

Mark-Up = 520 − 400 = ₱120 → Mark-Up % = (120/400) × 100 = 30%

📌 Example 6 – Discount %

₱2,000 reduced to ₱1,600. What is the discount percentage?

Discount = 400 → Discount % = (400/2000) × 100 = 20%

Topic 3 · Value-Added Tax (VAT)

Understanding VAT

VAT (Value-Added Tax) is a government-imposed tax on goods and services. In the Philippines, it is set at 12%. Businesses collect it from customers and remit it to the Bureau of Internal Revenue (BIR). Applied at Jollibee, SM, Lazada, Shopee, and more.

VAT Formulas

VAT = 12% × Price (VAT-exclusive)
Total (VAT-inclusive) = Price × 1.12
Price before VAT = Total ÷ 1.12

📌 Example 7 – VAT-Exclusive

Notebooks cost ₱500 (VAT-exclusive). VAT and total price?

VAT = 12% × 500 = ₱60 → Total = 500 + 60 = ₱560

📌 Example 8 – VAT from Inclusive Total

Jollibee bill is ₱224 (VAT-inclusive). How much is the VAT?

Price before VAT = 224 ÷ 1.12 = ₱200 → VAT = 224 − 200 = ₱24

📌 Example 9 – Price Before VAT

Shopee phone case costs ₱672 (VAT-inclusive). Price before VAT?

Price = 672 ÷ 1.12 = ₱600

📌 Example 10 – Adding VAT

Sari-sari store snacks have ₱1,000 base price + 12% VAT. Final price?

Total = 1,000 × 1.12 = ₱1,120

Topic 4 · Profit & Loss

Business Variables

Cost Price (CP)

Initial investment; price paid to buy or produce an item.

Selling Price (SP)

Final amount charged to customers.

Profit (Gain)

SP > CP → Profit = SP − CP

Loss

SP < CP → Loss = CP − SP

Profit %

Profit % = (Profit / CP) × 100

Loss %

Loss % = (Loss / CP) × 100

📌 Example 11 – Profit

Hoodie bought for ₱500, sold for ₱620. Profit and profit percentage?

Profit = 620 − 500 = ₱120 → Profit % = (120/500) × 100 = 24%

📌 Example 12 – Loss

Rice cooker bought at ₱1,800, sold at ₱1,500. Loss and loss percentage?

Loss = 1,800 − 1,500 = ₱300 → Loss % = (300/1800) × 100 ≈ 16.67%

📌 Example 13 – Target Selling Price

Phone bought for ₱10,000. You want 20% profit. What should be the selling price?

Profit = 20% × 10,000 = ₱2,000 → SP = 10,000 + 2,000 = ₱12,000

📌 Example 14 – Break-Even

Shoes bought for ₱1,200. Minimum selling price to avoid loss?

To avoid loss: SP = CP = ₱1,200

Must Memorize · Week 6

Inflation – general increase in prices over time; purchasing power of money decreases.
Mark-Up – amount added to CP to earn profit. SP = CP + Mark-Up
Discount – reduction from original price. Net Price = Original − Discount
Selling Price (SP) – final price charged to the customer after mark-up.
Net Price – actual amount paid after discount is applied.
VAT – government tax on goods/services; Philippines rate = 12%; collected by BIR.
VAT-Exclusive – price WITHOUT VAT. Total = Price × 1.12
VAT-Inclusive – price ALREADY has VAT. Original = Total ÷ 1.12
Profit – SP > CP. Formula: Profit = SP − CP. Profit % = (Profit/CP) × 100
Loss – SP < CP. Formula: Loss = CP − SP. Loss % = (Loss/CP) × 100
Gross Pay – salary before deductions.
Net Pay – salary after deductions (taxes, SSS, PhilHealth, Pag-IBIG).
SSS / GSIS – government-mandated deduction for employee retirement benefits.
Percentage Increase – [(New − Old) / Old] × 100
Percentage Decrease – [(Old − New) / Old] × 100

📝 Week 6

Possible Exam Questions

Percentages in Business & Finance

A Identification – 10 items

1.The term for salary before all deductions.
2.The term for salary after all deductions.
3.The government-mandated deduction intended for employee retirement benefits.
4.The general increase in prices of goods and services over time.
5.The government-imposed tax on goods and services in the Philippines set at 12%.
6.The amount added to the cost price to make a profit.
7.The actual amount a buyer pays after the discount has been deducted.
8.The government agency that collects VAT from businesses.
9.The business situation when Selling Price is less than Cost Price.
10.Formula to find the original price from a VAT-inclusive total.

B Multiple Choice – 10 items

1.
Which best describes “gross pay”?
- A. Salary before deductions
- B. Salary after deductions
- C. Additional benefits only
- D. Net income
2.
Which deduction is for retirement benefits?
- A. TAX
- B. PhilHealth
- C. Pag-IBIG
- D. GSIS/SSS
3.
Employee earns ₱20,000 with ₱5,000 deductions. Net pay?
- A. ₱25,000
- B. ₱20,000
- C. ₱15,000
- D. ₱5,000
4.
Which shows the effect of inflation?
- A. Cellphone price drops during a sale
- B. Grocery prices increase compared to last year
- C. A store gives a 10% discount
- D. A business earns more profit
5.
What does VAT represent?
- A. A discount given by the store
- B. A government-imposed tax on goods and services
- C. A company's profit margin
- D. An employee's retirement deduction
6.
Store buys a bag for ₱800, sells for ₱1,000. Mark-up percentage?
- A. 20%
- B. 25%
- C. 30%
- D. 15%
7.
Jacket priced ₱2,000 with 15% discount. Net price?
- A. ₱1,500
- B. ₱1,700
- C. ₱1,800
- D. ₱1,850
8.
Book costs ₱250 (VAT-exclusive). Total with 12% VAT?
- A. ₱262
- B. ₱275
- C. ₱280
- D. ₱300
9.
Goods bought ₱3,000, sold ₱2,700. This is an example of:
- A. Profit
- B. Mark-up
- C. Loss
- D. Break-even
10.
VAT-inclusive price is ₱560. Price before VAT?
- A. ₱480
- B. ₱490
- C. ₱500
- D. ₱520

C True or False – 5 items

1.The Philippines VAT rate is 10%.
2.Inflation causes the purchasing power of money to decrease.
3.Net pay is the salary before any deductions.
4.A business earns profit when the selling price is greater than the cost price.
5.To find the price before VAT from a VAT-inclusive total, divide the total by 1.12.

D Problem Solving – 5 items

1.The price of eggs increased from ₱7 to ₱9. Find the percentage increase.
2.A store buys shoes for ₱1,200 and sells them for ₱1,500. Find the mark-up percentage.
3.A bag costs ₱3,000. During a 20% sale, how much is the discounted price?
4.A restaurant bill before VAT is ₱1,200. What is the total with 12% VAT?
5.A trader bought fruits for ₱2,000 and sold them for ₱2,400. Find the profit and profit percentage.

🔑 Answer Key – Week 6

A. Identification

1.Gross Pay

2.Net Pay

3.GSIS / SSS

4.Inflation

5.VAT (Value-Added Tax)

6.Mark-Up

7.Net Price

8.BIR

9.Loss

10.Price = Total ÷ 1.12

B. Multiple Choice

1.A

2.D

3.C

4.B

5.B

6.B (25%)

7.B (₱1,700)

8.C (₱280)

9.C

10.C (₱500)

C. True or False

1.False (12%)

2.True

3.False (after)

4.True

5.True

D. Problem Solving

1.≈28.57% increase

2.25% mark-up

3.₱2,400 net price

4.₱1,344 total

5.₱400 profit; 20%

Week 7 · General Mathematics

Patterns in Life and Nature

Arithmetic · Geometric · Fibonacci · Harmonic · Perfect Squares · Constant Sequences

Learning Objectives

Define patterns, including the Fibonacci sequence, and identify patterns found in art, nature, and everyday life.
Determine the next terms of a given pattern and formulate its governing rule.
Create a visual showcase of at least two real-life patterns and present the mathematical rule behind each.
Appreciate the beauty, order, and importance of patterns in mathematics, nature, art, and everyday life.

Concept 1 · Patterns Around Us

What is a Pattern?

A pattern is a repeated or regular arrangement of shapes, numbers, or events. Patterns can be visual (spirals in shells, symmetry in flowers) or numerical (sequences that grow predictably). Patterns in nature emerge from physical laws and biological principles.

Fractals

Self-similar repeating forms in trees, rivers, lightning bolts, and blood vessels.

Fibonacci & Golden Ratio

Seen in sunflowers, pinecones, nautilus shells; optimize packing and growth. Golden Ratio ≈ 1.618.

Symmetry

Bilateral (left-right) or radial (all-direction) in animals and plants; aids in movement and camouflage.

Tessellations & Spirals

Beehives, spider webs, and shells showcase geometric efficiency in nature.

Rhythmic Patterns (Music)

Repetition and variation in beats give music its pulse; composers use Fibonacci numbers in composition.

Everyday Patterns

Coffee ring effects, cracked mud (desiccation), bubble clusters — all follow mathematical principles.

Concept 2 · Types of Number Sequences

Arithmetic Sequence

Arithmetic sequence — the difference between any two consecutive terms is constant (common difference d). Also called an arithmetic progression. Each term = previous term + d.

General Formula

aₙ = a₁ + (n − 1)d

aₙ = nth term

a₁ = first term

n = number of terms

d = common difference

📌 Example 1 – Arithmetic (d = 2)

Sequence: 2, 4, 6, 8, …

2468101214

Geometric Sequence

Geometric sequence — each term is found by multiplying the previous term by the common ratio r.

General Formula

aₙ = a₁ · rⁿ⁻¹

aₙ = nth term

a₁ = first term

n = number of terms

r = common ratio

📌 Example 2 – Geometric (r = 2)

Sequence: 3, 6, 12, 24, …

3612244896192

Fibonacci Sequence

Each term is the sum of the two preceding terms. No common difference or ratio. Starts: 0, 1, 1, 2, 3, 5, 8, 13, 21…

Recursive Formula

Fₙ = Fₙ₋¹ + Fₙ₋²

📌 Example 3 – Fibonacci

Sequence: 1, 1, 2, 3, 5, …

1123581321

📌 Application – Rabbit Population

A pair of rabbits produces a new pair each month from the 2nd month. How many pairs after 6 months? After 10 months?

Pairs: 1, 1, 2, 3, 5 → a₆ = 3+5 = 8 pairs
Continuing: a₇=13, a₈=21, a₉=34 → a₁₀ = 55 pairs

Harmonic Sequence

Harmonic sequence — the reciprocals of an arithmetic sequence. No constant difference or ratio.

If arithmetic is a, a+d, a+2d, …

Harmonic: 1/a, 1/(a+d), 1/(a+2d), …

📌 Example 4 – Harmonic

Arithmetic: 2, 3, 4, 5, … → Harmonic: 1/2, 1/3, 1/4, 1/5, …

1/21/31/41/51/61/71/8

Perfect Squares Sequence

Each term = n². No constant difference or ratio.

Formula

aₙ = n²

📌 Example 5 – Perfect Squares

14916253649

Constant Sequence

Every term is the same. Considered both arithmetic (d = 0) and geometric (r = 1).

Formula

aₙ = c (where c is the constant value)

📌 Example 6 – Constant

6666666

Concept 3 · How to Identify the Rule

Write down the sequence.
Check differences between consecutive terms. If constant → Arithmetic.
Check ratios between consecutive terms. If constant → Geometric.
If neither applies, check for Fibonacci (each term = sum of two previous) or other special rules.

📌 Real-Life: Weekly Savings

Save ₱50 every week. How much after 10 weeks? After 25 weeks?

a₁ = 50, d = 50 → aₙ = 50n
a₁₀ = 50(10) = ₱500 | a₂₅ = 50(25) = ₱1,250

📌 Real-Life: Viral Video

Video doubles views daily. Day 1 = 1,000 views. Views on Day 7? Day 10?

a₁ = 1,000, r = 2 → aₙ = 1,000 × 2ⁿ⁻¹
Day 7: 1,000 × 2⁴ = 64,000 | Day 10: 1,000 × 2⁹ = 512,000

Must Memorize · Week 7

Pattern – repeated or regular arrangement of shapes, numbers, or events.
Sequence – ordered list of numbers following a certain rule.
Arithmetic Sequence – constant difference d. Formula: aₙ = a₁ + (n−1)d
Common Difference (d) – constant added to each term in an arithmetic sequence.
Geometric Sequence – constant ratio r. Formula: aₙ = a₁ · rⁿ⁻¹
Common Ratio (r) – constant multiplied to each term in a geometric sequence.
Fibonacci Sequence – Fₙ = Fₙ₋¹ + Fₙ₋². Starts: 0, 1, 1, 2, 3, 5, 8, 13, 21…
Harmonic Sequence – reciprocals of an arithmetic sequence (1/a, 1/(a+d), …).
Perfect Squares – aₙ = n². Example: 1, 4, 9, 16, 25…
Constant Sequence – every term is the same; d = 0, r = 1.
Fractals – self-similar repeating patterns in trees, rivers, and lightning.
Symmetry – bilateral (left-right) or radial (all-direction) pattern.
Tessellation – shapes fitting together without gaps; e.g., beehives.
Golden Ratio – ≈ 1.618; related to Fibonacci; seen in art and nature.

📝 Week 7

Possible Exam Questions

Patterns in Life and Nature

A Identification – 10 items

1.Type of sequence where each term is the sum of the two preceding terms.
2.The constant added to each term in an arithmetic sequence.
3.The constant multiplied to each term in a geometric sequence.
4.Type of sequence formed by taking the reciprocals of an arithmetic sequence.
5.Self-similar repeating patterns found in trees, rivers, and lightning bolts.
6.General formula for the nth term of an arithmetic sequence.
7.Type of sequence where every term is the same (d = 0, r = 1).
8.Next term in the Fibonacci sequence after 21.
9.Formula for the sequence of perfect squares.
10.Type of symmetry where left and right sides mirror each other.

B Multiple Choice – 10 items

1.
Patterns in art and nature can be described by:
- A. Guesswork
- B. Careful inspection
- C. Random selection
- D. Trial and error
2.
Petal patterns in flowers are an example of:
- A. Random sequence
- B. Arithmetic sequence
- C. Geometric sequence
- D. Fibonacci sequence
3.
The Fibonacci sequence starts with:
- A. 0 and 1
- B. 1 and 1
- C. 2 and 3
- D. 1 and 2
4.
In the Fibonacci sequence, the next term after 21 is:
- A. 22
- B. 32
- C. 34
- D. 36
5.
The rule of a pattern explains:
- A. How each term is obtained
- B. The sum of all terms
- C. The random order of terms
- D. The largest term
6.
7th term of the sequence 2, 4, 6, 8, …?
- A. 12
- B. 14
- C. 16
- D. 18
7.
Which sequence is geometric?
- A. 2, 4, 6, 8
- B. 1, 1, 2, 3, 5
- C. 3, 6, 12, 24
- D. 1, 4, 9, 16
8.
a₁ = 500, d = 500. What is a₁₀?
- A. ₱4,500
- B. ₱5,000
- C. ₱5,500
- D. ₱4,000
9.
Viral video doubles daily from 1,000. Views on Day 7?
- A. 32,000
- B. 48,000
- C. 64,000
- D. 128,000
10.
Which pattern is found in beehives and spider webs?
- A. Fractals
- B. Fibonacci
- C. Tessellations and Spirals
- D. Symmetry

C True or False – 5 items

1.An arithmetic sequence is formed by multiplying each term by a constant ratio.
2.The Fibonacci sequence has no common difference or common ratio.
3.A constant sequence can be considered both arithmetic (d=0) and geometric (r=1).
4.The sequence 1, 4, 9, 16 is an arithmetic sequence.
5.Fractals are self-similar repeating patterns found in nature like trees and rivers.

D Short Answer / Problem Solving – 5 items

1.Find the next three terms of 5, 10, 20, 40, … and identify its type and common ratio.
2.Find the 10th term of the arithmetic sequence 3, 7, 11, 15, …
3.List the next three terms of the Fibonacci sequence: 1, 1, 2, 3, 5, 8, 13, …
4.A pair of rabbits produces another pair each month. After 6 months, how many pairs? (Use Fibonacci.)
5.Give two examples of patterns found in nature and explain the mathematical rule behind each.

🔑 Answer Key – Week 7

A. Identification

1.Fibonacci Sequence

2.Common Difference (d)

3.Common Ratio (r)

4.Harmonic Sequence

5.Fractals

6.aₙ = a₁ + (n−1)d

7.Constant Sequence

8.34

9.aₙ = n²

10.Bilateral Symmetry

B. Multiple Choice

1.B

2.D

3.A (or B)

4.C (34)

5.A

6.B (14)

7.C

8.B (5,000)

9.C (64,000)

10.C

C. True or False

1.False (geometric)

2.True

3.True

4.False (perfect squares)

5.True

D. Short Answer

1.80, 160, 320; Geometric, r=2

2.a₁₀ = 3+(9×4) = 39

3.21, 34, 55

4.8 pairs

5.e.g. Fibonacci in sunflowers; Fractals in trees

Week 8 · General Mathematics

Arithmetic & Geometric Sequences

Savings · Salary Increments · Installment Plans · Population Growth · Depreciation · Investment Growth

Concept 1 · Comparing the Two Sequences

Feature	Arithmetic Sequence	Geometric Sequence
Growth Pattern	Adds the same amount each step	Multiplies by the same ratio each step
Common Value	Difference d	Ratio r
Formula	`aₙ = a₁ + (n−1)d`	`aₙ = a₁ · rⁿ⁻¹`
Finance Example	Fixed savings, salary raises, installment payments	Compound interest, population growth, depreciation
Model Type	Linear / steady growth	Exponential growth or decay

Concept 2 · Arithmetic Sequence Applications

Savings & Salary

📌 Example 1 – Weekly Savings

Save ₱500/week. Sequence: 500, 1000, 1500, … How much after 10 weeks? After 25 weeks?

a₁ = 500, d = 500
After 10 weeks: a₁₀ = 500 + (10−1)(500) = ₱5,000
After 25 weeks: a₂₅ = 500 + (25−1)(500) = ₱12,500

📌 Example 2 – Salary Growth

Worker earns ₱12,000 in Month 1 with ₱500 monthly increase. In what month will he reach ₱15,000?

a₁ = 12,000, d = 500, aₙ = 15,000
15,000 = 12,000 + (n−1)(500) → 3,000 = (n−1)(500) → n−1 = 6 → n = 7 (7th month)

Installment Payments

📌 Example 3 – Gadget Purchase

Smartphone ₱18,000 on a 6-month installment plan of ₱3,000/month. Total paid on the 4th installment?

a₁ = 3,000, d = 3,000, n = 4
a₄ = 3,000 + (4−1)(3,000) = 3,000 + 9,000 = ₱12,000

📌 Example 4 – School Tuition

Student pays ₱2,500/month for 10 months. Total paid by the 10th month?

a₁ = 2,500, d = 2,500, n = 10
a₁₀ = 2,500 + (10−1)(2,500) = ₱25,000

📌 Example 5 – Car Loan

Car loan: ₱7,500/month for 36 months. Total paid after 2 years (24 months)?

a₁ = 7,500, d = 7,500, n = 24
a₂₄ = 7,500 + (24−1)(7,500) = 7,500 + 172,500 = ₱180,000

Concept 3 · Geometric Sequence Applications

Population Growth

📌 Example 6 – Bacteria

Culture starts with 100 bacteria, doubles every hour. How many after 5 hours?

a₁ = 100, r = 2, n = 5 → a₅ = 100 × 2⁴ = 100 × 16 = 1,600 bacteria

📌 Example 7 – Town Population

Town has 10,000 residents, growing at 5%/year. Population after 3 years?

a₁ = 10,000, r = 1.05, n = 3
a₃ = 10,000 × (1.05)² = 10,000 × 1.1025 = 11,025

📌 Example 8 – Startup Users

App gains 1,000 users in Week 1 and doubles each week. Users by Week 6?

a₁ = 1,000, r = 2, n = 6 → a₆ = 1,000 × 2⁵ = 1,000 × 32 = 32,000 users

Depreciation & Investment

📌 Example 9 – Laptop Depreciation

Laptop costs ₱60,000 and loses 15%/year. Value after 4 years?

a₁ = 60,000, r = 1−0.15 = 0.85, n = 4
a₄ = 60,000 × (0.85)³ = 60,000 × 0.614125 = ₱36,847.50

📌 Example 10 – TV Depreciation

TV worth ₱30,000 depreciates 20%/year. Value after 3 years?

a₁ = 30,000, r = 0.80, n = 3 → a₃ = 30,000 × (0.80)² = ₱19,200

📌 Example 11 – Investment Growth / Finding r

John invested ₱1,000. After one year it grew to ₱1,200. What is the common ratio?

r = a₂ ÷ a₁ = 1,200 ÷ 1,000 = r = 1.2 (20% growth per year)

Must Memorize · Week 8

Arithmetic Formula – aₙ = a₁ + (n−1)d where d = common difference
Geometric Formula – aₙ = a₁ · rⁿ⁻¹ where r = common ratio
Finding d – d = any term − previous term (arithmetic)
Finding r – r = any term ÷ previous term (geometric)
Depreciation Rate – r = 1 − rate. Example: 15% depreciation → r = 0.85
Growth Rate – r = 1 + rate. Example: 5% growth → r = 1.05
Arithmetic = Linear – steady, fixed increments (savings, salary, installments)
Geometric = Exponential – rapid growth or decay (bacteria, investments, depreciation)
Finding n – n = [(aₙ − a₁) ÷ d] + 1
Fibonacci (Review) – Fₙ = Fₙ₋¹ + Fₙ₋²; models natural population growth

📝 Week 8

Possible Exam Questions

Arithmetic & Geometric Sequences in Real Life

A Identification – 10 items

1.Formula for the nth term of an arithmetic sequence.
2.Formula for the nth term of a geometric sequence.
3.Type of sequence used to model fixed monthly installment payments.
4.Type of sequence used to model bacteria doubling every hour.
5.A laptop loses 15% of its value yearly. What is the common ratio?
6.The constant added in each term of an arithmetic sequence.
7.The constant multiplier in each term of a geometric sequence.
8.Type of growth modeled by a geometric sequence with r > 1.
9.Type of change modeled by a geometric sequence with 0 < r < 1.
10.Value of r when a town's population grows at 5% per year.

B Multiple Choice – 10 items

1.
Which situation best models an arithmetic sequence?
- A. Bacteria doubling every hour
- B. Saving ₱500 every week
- C. Population growing at 5% per year
- D. Gadget losing 15% yearly
2.
Worker earns ₱12,000 with ₱500 monthly increase. In what month will he earn ₱15,000?
- A. 5th month
- B. 6th month
- C. 7th month
- D. 8th month
3.
Save ₱500/week. How much after 10 weeks?
- A. ₱4,500
- B. ₱5,000
- C. ₱5,500
- D. ₱4,000
4.
100 bacteria double every hour. How many after 5 hours?
- A. 800
- B. 1,200
- C. 1,600
- D. 3,200
5.
Phone costs ₱20,000 and loses 10%/year. Value after 5 years?
- A. ₱10,000
- B. ₱11,809
- C. ₱12,500
- D. ₱13,122
6.
App starts with 1,000 users and doubles weekly. Users by Week 6?
- A. 16,000
- B. 32,000
- C. 64,000
- D. 48,000
7.
Car loan ₱7,500/month. Total paid after 24 months?
- A. ₱120,000
- B. ₱150,000
- C. ₱180,000
- D. ₱270,000
8.
In aₙ = a₁ · rⁿ⁻¹, what does “r” represent?
- A. Common difference
- B. Number of terms
- C. Common ratio
- D. First term
9.
Town of 10,000 grows 5%/year. Population after 3 years?
- A. 11,000
- B. 11,025
- C. 11,050
- D. 10,500
10.
Which correctly describes a geometric sequence?
- A. Terms increase by a fixed amount
- B. Terms are all the same
- C. Terms are multiplied by a constant ratio
- D. Terms are the sum of two previous terms

C True or False – 5 items

1.Fixed monthly savings is best modeled by a geometric sequence.
2.Depreciation of an asset can be modeled using a geometric sequence with r < 1.
3.In a geometric sequence with r = 2, each term is double the previous term.
4.The common difference in 500, 1000, 1500, 2000 is 500.
5.Arithmetic sequences model exponential growth patterns.

D Problem Solving – 5 items · Show complete solution & identify type

1.You save ₱500 every month. How much will you have after 12 months?
2.Population of rabbits starts at 50 and doubles every month. How many after 6 months?
3.A phone costs ₱20,000 and loses 10% of its value each year. What is its value after 5 years?
4.You pay a monthly installment of ₱2,000 for 18 months. How much have you paid after 10 months?
5.A small investment grows by 5% each month starting at ₱1,000. How much is it worth after 4 months?

🔑 Answer Key – Week 8

A. Identification

1.aₙ = a₁ + (n−1)d

2.aₙ = a₁ · rⁿ⁻¹

3.Arithmetic Sequence

4.Geometric Sequence

5.r = 0.85

6.Common Difference (d)

7.Common Ratio (r)

8.Exponential Growth

9.Decay / Decrease

10.r = 1.05

B. Multiple Choice

1.B

2.C (7th)

3.B (5,000)

4.C (1,600)

5.D (≈13,122)

6.B (32,000)

7.C (180,000)

8.C

9.B (11,025)

10.C

C. True or False

1.False (arithmetic)

2.True

3.True

4.True

5.False (geometric)

D. Problem Solving

1.Arithmetic; a₁₂ = ₱6,000

2.Geometric; a₆ = 1,600 rabbits

3.Geometric; a₅ ≈ ₱13,122

4.Arithmetic; a₁₀ = ₱20,000

5.Geometric; a₄ ≈ ₱1,157.63

⚡ One-Minute Final Review

General Mathematics · Weeks 6–8

Week 6: Percentages govern daily financial decisions. Inflation erodes buying power. Mark-ups and discounts shape prices. VAT = 12% government tax collected by BIR. Profit when SP > CP; Loss when SP < CP.

Week 7: Patterns are everywhere — in nature (Fibonacci spirals, fractals, symmetry), in music, and in mathematics. Arithmetic sequences grow by adding d; geometric sequences grow by multiplying r; Fibonacci sequences add the two previous terms.

Week 8: Arithmetic sequences model steady linear growth (savings, salaries, installments). Geometric sequences model exponential growth or decay (bacteria, investments, depreciation). Identify the type, apply the formula, and compute.

💰

Week 6 Key Formulas

% Change = [(New−Old)/Old]×100
SP = CP + Mark-Up
Net Price = Original − Discount
VAT = 12% × Price
VAT Total = Price × 1.12
Profit = SP−CP | Loss = CP−SP

🌿

Week 7 Key Sequences

Arithmetic: aₙ = a₁ + (n−1)d
Geometric: aₙ = a₁ · rⁿ⁻¹
Fibonacci: Fₙ = Fₙ₋¹ + Fₙ₋²
Harmonic: 1/a, 1/(a+d), …
Perfect Squares: aₙ = n²
Constant: aₙ = c

📈

Week 8 Applications

Savings/Salary/Installments → Arithmetic
Growth rate: r = 1 + rate
Decay rate: r = 1 − rate
Bacteria/Population → Geometric
Depreciation → Geometric (r < 1)
Find n: n = [(aₙ−a₁)/d] + 1