Algebra 1 (HS pathway)

Functions and Sequences

Algebra 1

✓By the end of this lesson students will be able to understand and use function notation to represent mathematical relationships.
✓By the end of this lesson students will be able to identify and distinguish between arithmetic and geometric sequences.
✓By the end of this lesson students will be able to write explicit formulas for arithmetic sequences.
✓By the end of this lesson students will be able to write explicit formulas for geometric sequences.
✓By the end of this lesson students will be able to use explicit formulas to find specific terms in arithmetic and geometric sequences.

Key concepts

Function Notation

Function notation is a way to represent functions algebraically. Instead of using 'y' for the dependent variable, we use 'f(x)', which is read as 'f of x' or 'the value of f at x'. The 'x' inside the parentheses represents the input value, and 'f(x)' represents the output value corresponding to that input. This notation clearly indicates that the output depends on the input 'x'.

y = f(x)

Arithmetic Sequences

An arithmetic sequence is a sequence of numbers such that the difference between consecutive terms is constant. This constant difference is called the common difference, denoted by 'd'. Each term after the first is found by adding the common difference to the previous term.

a_n = a_1 + (n-1)d

Geometric Sequences

A geometric sequence is a sequence of numbers such that the ratio of consecutive terms is constant. This constant ratio is called the common ratio, denoted by 'r'. Each term after the first is found by multiplying the previous term by the common ratio.

a_n = a_1 * r^(n-1)

Key facts to remember

1Function notation f(x) represents the output value of a function for a given input x.
2An arithmetic sequence has a constant common difference (d) between consecutive terms.
3The explicit formula for an arithmetic sequence is a_n = a_1 + (n-1)d.
4A geometric sequence has a constant common ratio (r) between consecutive terms.
5The explicit formula for a geometric sequence is a_n = a_1 * r^(n-1).
6a_1 represents the first term of a sequence.
7a_n represents the nth term of a sequence.
8n represents the term number (position) in a sequence.

Worked examples

Example 1

Given the function f(x) = 4x - 7, find f(5).

ISubstitute the input value x = 5 into the function's rule.

IIf(5) = 4(5) - 7

IIIPerform the multiplication.

IVf(5) = 20 - 7

VPerform the subtraction.

VIf(5) = 13

Answer

f(5) = 13

Remember that f(5) does not mean f multiplied by 5; it means the value of the function f when the input is 5.

Example 2

For the arithmetic sequence 5, 11, 17, 23, ..., find the explicit formula and the 15th term.

IIdentify the first term (a_1) and the common difference (d).

IIa_1 = 5

IIId = 11 - 5 = 6 (or 17 - 11 = 6)

IVWrite the explicit formula for an arithmetic sequence: a_n = a_1 + (n-1)d.

VSubstitute a_1 = 5 and d = 6 into the formula.

VIa_n = 5 + (n-1)6

VIIDistribute the 6.

VIIIa_n = 5 + 6n - 6

9Combine like terms to simplify the formula.

10a_n = 6n - 1

11To find the 15th term, substitute n = 15 into the explicit formula.

12a_15 = 6(15) - 1

13a_15 = 90 - 1

14a_15 = 89

Answer

Explicit formula: a_n = 6n - 1. The 15th term is a_15 = 89.

Always simplify the explicit formula to its most concise form after substituting a_1 and d.

Example 3

For the geometric sequence 3, 6, 12, 24, ..., find the explicit formula and the 7th term.

IIdentify the first term (a_1) and the common ratio (r).

IIa_1 = 3

IIIr = 6 / 3 = 2 (or 12 / 6 = 2)

IVWrite the explicit formula for a geometric sequence: a_n = a_1 * r^(n-1).

VSubstitute a_1 = 3 and r = 2 into the formula.

VIa_n = 3 * 2^(n-1)

VIITo find the 7th term, substitute n = 7 into the explicit formula.

VIIIa_7 = 3 * 2^(7-1)

9a_7 = 3 * 2^6

10Calculate 2^6.

11a_7 = 3 * 64

12Perform the multiplication.

13a_7 = 192

Answer

Explicit formula: a_n = 3 * 2^(n-1). The 7th term is a_7 = 192.

Be careful with the order of operations: calculate the exponent (n-1) first, then the power, and finally multiply by a_1.

Common mistakes

✗Confusing f(x) as 'f times x' instead of 'f of x' or 'the value of f at x'.
✗Mixing up the formulas for arithmetic and geometric sequences.
✗Incorrectly calculating the common difference (d) or common ratio (r). For example, dividing instead of subtracting for 'd', or subtracting instead of dividing for 'r'.
✗Errors in order of operations when evaluating geometric sequences, especially with the exponent (n-1).
✗Forgetting to simplify the arithmetic sequence formula after substituting a_1 and d.

Exam tips

★Always clearly identify the first term (a_1), common difference (d), or common ratio (r) before writing the formula.
★Show all steps when evaluating functions or finding specific terms in sequences to minimize calculation errors.
★Double-check your calculations, especially when dealing with negative numbers or exponents.
★Pay close attention to the wording of the question to determine if it's asking for an explicit formula, a specific term, or to identify the type of sequence.

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