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Algebra & Functions

Algebra & Number Patterns

Grade 10 · Grade 11 · Grade 12

  • ✓By the end of this lesson students will be able to solve various types of equations and inequalities, including linear, quadratic, and exponential forms.
  • ✓By the end of this lesson students will be able to identify and describe different types of number patterns, specifically arithmetic, geometric, and quadratic sequences.
  • ✓By the end of this lesson students will be able to determine the general term (nth term) of arithmetic, geometric, and quadratic sequences.
  • ✓By the end of this lesson students will be able to calculate the sum of arithmetic and geometric series, including the sum to infinity for converging geometric series.
  • ✓By the end of this lesson students will be able to apply their knowledge of equations, inequalities, and number patterns to solve contextual problems.

Key concepts

Equations

An equation is a mathematical statement that asserts the equality of two expressions. Solving an equation means finding the value(s) of the variable(s) that make the statement true.

Linear Equations

Equations where the highest power of the variable is 1. They can be solved by isolating the variable.

ax + b = c
Quadratic Equations

Equations where the highest power of the variable is 2. They can be solved by factorisation, using the quadratic formula, or completing the square.

ax^2 + bx + c = 0
Simultaneous Equations

A set of two or more equations with two or more variables that must be solved together to find values that satisfy all equations in the set. Methods include substitution and elimination.

Exponential Equations

Equations where the variable appears in the exponent. They are often solved by expressing both sides with the same base.

a^x = b
Inequalities

A mathematical statement comparing two expressions using inequality symbols (<, >, ≤, ≥). Solving an inequality means finding the range of values for the variable(s) that make the statement true.

Linear Inequalities

Inequalities where the highest power of the variable is 1. Solved similarly to linear equations, but remember to reverse the inequality sign if multiplying or dividing by a negative number.

ax + b > c
Quadratic Inequalities

Inequalities where the highest power of the variable is 2. Solved by finding critical values (roots of the corresponding quadratic equation) and then using a number line or sketch to determine the solution intervals.

ax^2 + bx + c > 0
Sequence

An ordered list of numbers, called terms, that follow a specific pattern or rule.

Series

The sum of the terms of a sequence.

Arithmetic Sequence

A sequence where the difference between consecutive terms is constant. This constant difference is called the common difference (d).

T_n = a + (n-1)d
Arithmetic Series

The sum of the terms of an arithmetic sequence.

S_n = n/2 [2a + (n-1)d] OR S_n = n/2 [a + L]
Geometric Sequence

A sequence where the ratio between consecutive terms is constant. This constant ratio is called the common ratio (r).

T_n = ar^(n-1)
Geometric Series

The sum of the terms of a geometric sequence.

S_n = a(r^n - 1) / (r - 1) (for r ≠ 1)
Converging Geometric Series (Sum to Infinity)

A geometric series converges to a finite sum if the common ratio 'r' is between -1 and 1 (i.e., -1 < r < 1).

S_∞ = a / (1 - r) (for -1 < r < 1)
Quadratic Sequence

A sequence where the second differences between consecutive terms are constant. The general term is a quadratic expression in terms of 'n'.

T_n = an^2 + bn + c

Key facts to remember

  • 1The general term for an arithmetic sequence is T_n = a + (n-1)d.
  • 2The general term for a geometric sequence is T_n = ar^(n-1).
  • 3For a quadratic sequence T_n = an^2 + bn + c, the second difference is equal to 2a.
  • 4The sum of an arithmetic series is S_n = n/2 [2a + (n-1)d] or S_n = n/2 [a + L].
  • 5The sum of a geometric series is S_n = a(r^n - 1) / (r - 1) (for r ≠ 1).
  • 6A geometric series converges to a sum to infinity, S_∞ = a / (1 - r), only if -1 < r < 1.
  • 7When solving inequalities, remember to reverse the inequality sign if you multiply or divide by a negative number.
  • 8The discriminant Δ = b^2 - 4ac of a quadratic equation ax^2 + bx + c = 0 determines the nature of its roots.

Worked examples

Example 1

Solve for x: x^2 - 5x + 6 ≤ 0

IStep 1: Convert the inequality to an equation to find the critical values.
IIx^2 - 5x + 6 = 0
IIIStep 2: Factorise the quadratic equation.
IV(x - 2)(x - 3) = 0
VStep 3: Determine the critical values.
VIx - 2 = 0 OR x - 3 = 0
VIIx = 2 OR x = 3
VIIIStep 4: Use a number line or sketch of the parabola y = x^2 - 5x + 6 to determine the interval where the expression is less than or equal to zero. The parabola opens upwards and intersects the x-axis at x=2 and x=3. The function is less than or equal to zero between and including these roots.
9Step 5: Write the solution in interval notation or set builder notation.

Answer

x ∈ [2; 3] OR 2 ≤ x ≤ 3

Remember to include the critical values in the solution if the inequality includes 'equal to' (≤ or ≥).

Example 2

Given the quadratic sequence 2; 5; 10; 17; ..., determine the general term T_n.

IStep 1: Find the first differences between consecutive terms.
II5 - 2 = 3
III10 - 5 = 5
IV17 - 10 = 7
VFirst differences: 3; 5; 7
VIStep 2: Find the second differences between the first differences.
VII5 - 3 = 2
VIII7 - 5 = 2
9Second differences: 2 (constant, confirming it's a quadratic sequence)
10Step 3: Use the relationships for a quadratic sequence T_n = an^2 + bn + c:
112a = second difference
123a + b = T_2 - T_1 (first term of the first differences)
13a + b + c = T_1 (first term of the sequence)
14Step 4: Solve for 'a'.
152a = 2
16a = 1
17Step 5: Solve for 'b' using 'a' and the first term of the first differences.
183(1) + b = 3
193 + b = 3
20b = 0
21Step 6: Solve for 'c' using 'a', 'b', and the first term of the sequence.
221 + 0 + c = 2
23c = 1
24Step 7: Substitute 'a', 'b', and 'c' into the general formula T_n = an^2 + bn + c.

Answer

T_n = n^2 + 1

Always check your general term by substituting n=1, n=2, etc., to ensure it generates the given sequence.

Example 3

Calculate the sum of the first 10 terms of the geometric series 2 + 6 + 18 + ...

IStep 1: Identify the first term (a) and the common ratio (r).
IIa = 2
IIIr = T_2 / T_1 = 6 / 2 = 3
IVStep 2: Identify the number of terms (n).
Vn = 10
VIStep 3: Write down the formula for the sum of a geometric series.
VIIS_n = a(r^n - 1) / (r - 1)
VIIIStep 4: Substitute the values into the formula.
9S_10 = 2(3^10 - 1) / (3 - 1)
10S_10 = 2(59049 - 1) / 2
11S_10 = 59048

Answer

S_10 = 59048

Ensure your calculator is used correctly for powers and order of operations.

Common mistakes

  • ✗Forgetting to reverse the inequality sign when multiplying or dividing by a negative number.
  • ✗Confusing the formulas for arithmetic and geometric sequences/series.
  • ✗Incorrectly calculating common differences or common ratios.
  • ✗Not checking the condition (-1 < r < 1) before attempting to calculate the sum to infinity.
  • ✗Errors in factorising quadratic expressions or applying the quadratic formula incorrectly.

Exam tips

  • ★Always show all your working steps clearly, as marks are often awarded for method even if the final answer is incorrect.
  • ★Read the question carefully to determine whether a sequence (T_n) or a series (S_n) is required.
  • ★For inequalities, always determine the critical values first and then use a number line or sketch to find the correct solution set.
  • ★Practise identifying the type of sequence (arithmetic, geometric, or quadratic) before attempting to apply any formulas.

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