Difficulty: Introductory to Intermediate | Prerequisites: Basic algebra, exponent rules, familiarity with the unit circle
This set of problems covers four distinct but related areas of pre-calculus and algebra: verifying algebraic identities, interpreting radian measure on the unit circle, converting recursive sequences to explicit (closed-form) formulas, and modelling compound interest with exponential expressions. These are foundational tools. Algebraic identities appear constantly in simplification and proof work. Radian measure is essential for trigonometry and calculus. Sequences and exponential growth underpin everything from population models to financial mathematics.
Know how to expand and verify polynomial identities, recognise what one radian looks like on a unit circle, convert a recursive geometric sequence to its explicit formula, and set up a compound interest expression with the correct base and exponent.
Algebraic identity
An equation that holds true for all permissible values of its variables. Think of it as a universal shortcut: it works no matter what numbers you plug in.
Binomial expansion
The result of raising a binomial (two-term expression) to a power. For a square: (a + b)² = a² + 2ab + b². For a cube: (a + b)³ = a³ + 3a²b + 3ab² + b³. In simple terms, it tells you every term you get when you multiply the binomial by itself.
Pythagorean identity (for integers)
The relationship (a² + b²)² = (a² − b²)² + (2ab)², which generates Pythagorean triples. Think of it as a machine: feed in any two integers and it produces three numbers that satisfy a² + b² = c².
Radian
A unit of angle measure defined so that an arc length equal to the radius of the circle subtends an angle of one radian. One full revolution = 2π radians ≈ 6.28 radians. In simple terms, one radian is the angle you get when you wrap the radius along the circumference.
Recursive formula (recurrence relation)
A formula that defines each term using one or more previous terms, plus initial condition(s). For example, a₁ = 6, aₙ = 3aₙ₋₁.
Explicit (closed-form) formula
A formula that gives the nth term directly, without needing to compute earlier terms. For example, aₙ = 2 · 3ⁿ.
Compound interest
Interest calculated on both the initial principal and the accumulated interest from previous periods. The general formula is A = P(1 + r)ⁿ, where P is the principal, r is the rate per period, and n is the number of periods.
The question asks which of three rules hold for all real numbers.
Rule I: (m + p)² = m² + 2mp + p²
Expand (m + p)(m + p) = m² + mp + pm + p² = m² + 2mp + p².
This is the standard perfect-square trinomial. It is true for all real numbers.
Rule II: (x + y)³ = x³ + 3xy + y³
The correct expansion is (x + y)³ = x³ + 3x²y + 3xy² + y³.
Rule II is missing the 3x²y and 3xy² terms, replacing them with just 3xy. It is false.
Quick disproof: let x = 1, y = 1. Left side = 8. Right side = 1 + 3 + 1 = 5.
Rule III: (a² + b²)² = (a² − b²)² + (2ab)²
Expand the right side: (a² − b²)² + (2ab)² = a⁴ − 2a²b² + b⁴ + 4a²b² = a⁴ + 2a²b² + b⁴.
Expand the left side: (a² + b²)² = a⁴ + 2a²b² + b⁴.
Both sides match. This is true, and it is the identity behind Pythagorean triple generation.
Answer: (4) I and III
One radian is approximately 57.3°, which is slightly less than a 60° angle.
On the unit circle (radius = 1), one radian subtends an arc whose length equals 1 (the radius).
The correct diagram shows a counterclockwise rotation from the positive x-axis that lands in the first quadrant, roughly one-sixth of the way around the circle.
Answer: Diagram (1) shows the angle terminating in the first quadrant with an arc length equal to the radius, which is what one radian looks like.
The sequence is defined by a₁ = 6, aₙ = 3aₙ₋₁.
This is a geometric sequence with first term 6 and common ratio 3.
Write out a few terms to spot the pattern:
a₁ = 6
a₂ = 3 · 6 = 18
a₃ = 3 · 18 = 54
a₄ = 3 · 54 = 162
The general geometric sequence formula is aₙ = a₁ · rⁿ⁻¹ = 6 · 3ⁿ⁻¹.
Simplify: 6 · 3ⁿ⁻¹ = 6 · (3ⁿ / 3) = 2 · 3ⁿ.
Check: a₁ = 2 · 3¹ = 6. a₂ = 2 · 3² = 18. Correct.
Answer: (3) aₙ = 2 · 3ⁿ
Remi borrows $300 at 30% interest compounded every 14 days, for one year.
The compound interest formula is A = P(1 + r)ⁿ.
P = 300, r = 0.30 per period.
The growth factor per 14-day period is (1 + 0.30) = 1.30.
Number of 14-day periods in one year: 365 / 14.
So A = 300(1.30)^(365/14).
Answer: (4) 300(1.30)^(365/14)
Perfect square trinomial: (a + b)² = a² + 2ab + b²
Binomial cube: (a + b)³ = a³ + 3a²b + 3ab² + b³
Pythagorean identity: (a² + b²)² = (a² − b²)² + (2ab)²
Radian-degree conversion: 1 radian = 180/π ≈ 57.3°
Geometric sequence (explicit): aₙ = a₁ · rⁿ⁻¹
Compound interest: A = P(1 + r)ⁿ
Compound interest is the foundation of loan amortisation, savings growth, and credit card debt calculations. The payday loan in Q7 is a sharp illustration: 30% every two weeks sounds manageable until you realise the exponent (365/14 ≈ 26 periods) makes it explosive. Pythagorean triple generation, meanwhile, appears in computer graphics, cryptography, and number theory.
Students often assume (x + y)³ expands as simply as (x + y)², just with cubes. It does not: the cube has four terms with binomial coefficients 1, 3, 3, 1, not two middle terms.
A radian is not "about 90°." One radian ≈ 57.3°, so it sits well inside the first quadrant. Confusing radians with quarter-turns is a common error.
When converting a recursive geometric sequence to explicit form, students sometimes write aₙ = a₁ · rⁿ instead of aₙ = a₁ · rⁿ⁻¹. The exponent is (n − 1) because the first multiplication happens at the second term.
For compound interest, students sometimes use the rate alone (0.30) as the base instead of (1 + rate) = 1.30. The base must include the original principal, so it is always (1 + r).
⚠️ Algebraic identity verification is a standard proof/disproof question. Always expand both sides fully rather than eyeballing.
⚠️ Radian measure questions test whether you know the size of one radian. Memorise: 1 rad ≈ 57.3°, roughly the angle of a pizza slice from a pizza cut into just over six slices.
⚠️ Recursive-to-explicit conversion is a high-frequency exam item. Write out three or four terms, spot the pattern, then verify the formula at n = 1 and n = 2.
⚠️ Compound interest problems test whether you set up the exponent correctly. The exponent is the number of compounding periods, not the number of days or years.
True or False: (a + b)² = a² + b².
Fill in the blank: One radian is approximately ______ degrees.
True or False: The sequence a₁ = 6, aₙ = 3aₙ₋₁ has a common ratio of 6.
Fill in the blank: In the compound interest formula A = P(1 + r)ⁿ, the base (1 + r) for a 30% rate per period is ______.
True or False: (a² + b²)² = (a² − b²)² + (2ab)² is true for all real numbers a and b.
Answers: 1. False, you are missing the 2ab cross term. 2. 57.3°. 3. False, the common ratio is 3. 4. 1.30. 5. True.
Q: Which of the following is true for all real numbers: (m + p)² = m² + 2mp + p²?
A: Yes. Expanding (m + p)(m + p) gives m² + mp + pm + p² = m² + 2mp + p².
Q: Disprove (x + y)³ = x³ + 3xy + y³.
A: Let x = 1, y = 1. The left side is (1 + 1)³ = 8. The right side is 1 + 3(1)(1) + 1 = 5. Since 8 ≠ 5, the identity is false.
Q: Convert the recursive sequence a₁ = 6, aₙ = 3aₙ₋₁ to an explicit formula.
A: This is geometric with a₁ = 6, r = 3. So aₙ = 6 · 3ⁿ⁻¹ = 2 · 3ⁿ.
Q: A $300 loan compounds at 30% every 14 days. Write the expression for the amount owed after one year with no payments.
A: A = 300(1.30)^(365/14). The base is 1.30 (principal plus 30% interest) and the exponent is the number of 14-day periods in 365 days.
Q: What does an angle of one radian look like on the unit circle?
A: It is the angle formed when the arc length equals the radius. On the unit circle, this lands in the first quadrant, about 57.3° from the positive x-axis.
Binomial expansion connects to the Binomial Theorem and Pascal's Triangle, both of which generalise to higher powers and appear throughout combinatorics. Compound interest is a special case of exponential growth, which links to logarithms (when solving for time) and to continuous compounding via the constant e. Radian measure is the default unit in calculus, where derivatives of trig functions only work cleanly in radians.
algebraic identity, perfect square trinomial, binomial expansion, binomial theorem, Pascal's Triangle, Pythagorean triple, radian, degree-radian conversion, unit circle, arc length, geometric sequence, recursive formula, explicit formula, closed-form, common ratio, compound interest, exponential growth, payday loan math, compounding periods, growth factor