1. What is a Sequence?

Definition: A sequence is an ordered list of numbers that follows a specific pattern or rule. Each number in a sequence is called a term.

Notation:

• \( a_1 \) = first term
• \( a_2 \) = second term
• \( a_3 \) = third term
• \( a_n \) = nth term (general term)

Types of Sequences:

• Arithmetic Sequence: Terms differ by a constant amount (common difference)
• Geometric Sequence: Each term is found by multiplying by a constant (common ratio)

Example:

Sequence: 3, 7, 11, 15, 19, ...

\( a_1 = 3 \), \( a_2 = 7 \), \( a_3 = 11 \), \( a_4 = 15 \), \( a_5 = 19 \)

2. Arithmetic Sequences

Definition: An arithmetic sequence is a sequence where each term is found by adding a constant value (common difference) to the previous term.

Common Difference (d):

\( d = a_2 - a_1 = a_3 - a_2 = a_n - a_{n-1} \)

The common difference is the same between any two consecutive terms.

Nth Term Formula:

\( a_n = a_1 + (n - 1)d \)

where \( a_n \) = nth term, \( a_1 \) = first term, \( n \) = term number, \( d \) = common difference

Examples:

Example 1: Find the 10th term of the sequence: 5, 9, 13, 17, 21, ...

Step 1: Find the common difference: \( d = 9 - 5 = 4 \)

Step 2: Identify first term: \( a_1 = 5 \)

Step 3: Use formula: \( a_{10} = 5 + (10 - 1)(4) \)

\( a_{10} = 5 + 9(4) = 5 + 36 = 41 \)

Answer: The 10th term is 41

Example 2: Find the 20th term of: 100, 93, 86, 79, ...

\( d = 93 - 100 = -7 \) (negative common difference means decreasing)

\( a_{20} = 100 + (20 - 1)(-7) = 100 + 19(-7) = 100 - 133 = -33 \)

Example 3: The 5th term of an arithmetic sequence is 17 and the common difference is 3. Find the first term.

Using \( a_n = a_1 + (n-1)d \):

\( 17 = a_1 + (5-1)(3) \)

\( 17 = a_1 + 12 \) → \( a_1 = 5 \)

3. Geometric Sequences

Definition: A geometric sequence is a sequence where each term is found by multiplying the previous term by a constant value (common ratio).

Common Ratio (r):

\( r = \frac{a_2}{a_1} = \frac{a_3}{a_2} = \frac{a_n}{a_{n-1}} \)

The common ratio is found by dividing any term by the previous term.

Nth Term Formula:

\( a_n = a_1 \cdot r^{n-1} \)

where \( a_n \) = nth term, \( a_1 \) = first term, \( r \) = common ratio, \( n \) = term number

Examples:

Example 1: Find the 6th term of the sequence: 3, 6, 12, 24, 48, ...

Step 1: Find the common ratio: \( r = \frac{6}{3} = 2 \)

Step 2: Identify first term: \( a_1 = 3 \)

Step 3: Use formula: \( a_6 = 3 \cdot 2^{6-1} = 3 \cdot 2^5 \)

\( a_6 = 3 \cdot 32 = 96 \)

Answer: The 6th term is 96

Example 2: Find the 5th term of: 80, 40, 20, 10, ...

\( r = \frac{40}{80} = \frac{1}{2} = 0.5 \) (ratio less than 1 means decreasing)

\( a_5 = 80 \cdot \left(\frac{1}{2}\right)^{5-1} = 80 \cdot \left(\frac{1}{2}\right)^4 = 80 \cdot \frac{1}{16} = 5 \)

Example 3: Find the 7th term of: 2, -6, 18, -54, ...

\( r = \frac{-6}{2} = -3 \) (negative ratio means alternating signs)

\( a_7 = 2 \cdot (-3)^{7-1} = 2 \cdot (-3)^6 = 2 \cdot 729 = 1458 \)

4. Identify Arithmetic and Geometric Sequences

Steps to Identify:

1. Check for Arithmetic: Subtract consecutive terms. If the difference is constant → Arithmetic
2. Check for Geometric: Divide consecutive terms. If the ratio is constant → Geometric
3. Neither: If neither difference nor ratio is constant → Neither

Comparison Table:

Practice Examples:

Example 1: Identify the sequence: 7, 14, 21, 28, 35, ...

Check differences: 14-7=7, 21-14=7, 28-21=7

✓ Arithmetic with d = 7

Example 2: Identify the sequence: 5, 15, 45, 135, ...

Check ratios: 15÷5=3, 45÷15=3, 135÷45=3

✓ Geometric with r = 3

Example 3: Identify the sequence: 1, 4, 9, 16, 25, ...

Check differences: 4-1=3, 9-4=5, 16-9=7 (not constant)

Check ratios: 4÷1=4, 9÷4=2.25 (not constant)

✗ Neither (This is a sequence of perfect squares: 1², 2², 3², 4², 5²)

5. Sequences: Word Problems

Steps to Solve:

1. Read carefully and identify what's being asked
2. Determine if it's arithmetic (adding/subtracting) or geometric (multiplying/dividing)
3. Identify the first term (\( a_1 \))
4. Find the common difference (d) or common ratio (r)
5. Use the appropriate formula
6. Solve and interpret your answer with units

Arithmetic Word Problem Examples:

Example 1: A theater has 20 seats in the first row, 24 seats in the second row, 28 seats in the third row, and so on. How many seats are in the 15th row?

Type: Arithmetic (each row adds 4 seats)

\( a_1 = 20 \), \( d = 4 \), \( n = 15 \)

\( a_{15} = 20 + (15-1)(4) = 20 + 56 = 76 \)

Answer: The 15th row has 76 seats.

Example 2: A savings plan starts with $500 and increases by $75 each month. How much money is saved after 12 months?

\( a_1 = 500 \), \( d = 75 \), \( n = 12 \)

\( a_{12} = 500 + (12-1)(75) = 500 + 825 = 1325 \)

Answer: $1,325 after 12 months.

Geometric Word Problem Examples:

Example 3: A bacteria culture doubles every hour. If there are initially 50 bacteria, how many will there be after 6 hours?

Type: Geometric (doubling means r = 2)

\( a_1 = 50 \), \( r = 2 \), \( n = 7 \) (initial + 6 hours)

\( a_7 = 50 \cdot 2^{7-1} = 50 \cdot 2^6 = 50 \cdot 64 = 3200 \)

Answer: 3,200 bacteria after 6 hours.

Example 4: A ball bounces to half its previous height each time. If dropped from 80 feet, how high does it bounce on the 5th bounce?

\( a_1 = 80 \), \( r = \frac{1}{2} \), \( n = 5 \)

\( a_5 = 80 \cdot \left(\frac{1}{2}\right)^{5-1} = 80 \cdot \frac{1}{16} = 5 \)

Answer: 5 feet on the 5th bounce.

6. Evaluate Variable Expressions for Sequences

Skill: Given a formula for a sequence, find specific term values by substituting the term number.

Common Variable Expressions:

• Arithmetic: \( a_n = a_1 + (n-1)d \) or \( a_n = dn + c \)
• Geometric: \( a_n = a_1 \cdot r^{n-1} \)
• General: Any expression with variable n

Steps:

1. Identify the variable expression
2. Substitute the given value of n
3. Simplify using order of operations
4. Write the answer

Examples:

Example 1: A sequence is defined by \( a_n = 4n - 1 \). Find \( a_1 \), \( a_5 \), and \( a_{10} \).

\( a_1 \): \( a_1 = 4(1) - 1 = 4 - 1 = 3 \)

\( a_5 \): \( a_5 = 4(5) - 1 = 20 - 1 = 19 \)

\( a_{10} \): \( a_{10} = 4(10) - 1 = 40 - 1 = 39 \)

Example 2: A sequence is defined by \( a_n = 3 \cdot 2^{n-1} \). Find the first 4 terms.

\( a_1 = 3 \cdot 2^{1-1} = 3 \cdot 2^0 = 3 \cdot 1 = 3 \)

\( a_2 = 3 \cdot 2^{2-1} = 3 \cdot 2^1 = 3 \cdot 2 = 6 \)

\( a_3 = 3 \cdot 2^{3-1} = 3 \cdot 2^2 = 3 \cdot 4 = 12 \)

\( a_4 = 3 \cdot 2^{4-1} = 3 \cdot 2^3 = 3 \cdot 8 = 24 \)

Sequence: 3, 6, 12, 24 (geometric with r = 2)

Example 3: Given \( a_n = n^2 + 2 \), find \( a_6 \).

\( a_6 = 6^2 + 2 = 36 + 2 = 38 \)

7. Write Variable Expressions for Arithmetic Sequences

Goal: Create a formula that describes the nth term of an arithmetic sequence.

Two Forms of Arithmetic Expressions:

Form 1: Standard Arithmetic Formula

\( a_n = a_1 + (n-1)d \)

Form 2: Simplified Linear Form

\( a_n = dn + c \)

where \( d \) = common difference and \( c = a_1 - d \)

Steps to Write the Expression:

1. Find the common difference (d): subtract consecutive terms
2. Identify the first term (\( a_1 \))
3. Method 1: Substitute into \( a_n = a_1 + (n-1)d \)
4. Method 2: Expand and simplify to \( a_n = dn + c \)

Examples:

Example 1: Write an expression for the sequence: 7, 11, 15, 19, 23, ...

Step 1: Find d: \( 11 - 7 = 4 \)

Step 2: First term: \( a_1 = 7 \)

Step 3 (Method 1): \( a_n = 7 + (n-1)(4) \)

Expand: \( a_n = 7 + 4n - 4 = 4n + 3 \)

Expression: \( a_n = 4n + 3 \)

Example 2: Write an expression for: 50, 45, 40, 35, 30, ...

\( d = 45 - 50 = -5 \), \( a_1 = 50 \)

\( a_n = 50 + (n-1)(-5) \)

\( a_n = 50 - 5n + 5 = -5n + 55 \)

Expression: \( a_n = -5n + 55 \)

Example 3: The third term of an arithmetic sequence is 14 and the common difference is 3. Write the expression for \( a_n \).

\( a_3 = 14 \), \( d = 3 \)

Find \( a_1 \): \( 14 = a_1 + (3-1)(3) \) → \( 14 = a_1 + 6 \) → \( a_1 = 8 \)

\( a_n = 8 + (n-1)(3) = 8 + 3n - 3 = 3n + 5 \)

Expression: \( a_n = 3n + 5 \)

8. Sequences: Mixed Review

Key Skills to Master:

• Identify whether a sequence is arithmetic, geometric, or neither
• Find the common difference or common ratio
• Calculate specific terms using formulas
• Write variable expressions for sequences
• Solve real-world problems involving sequences

Practice Problems:

Problem 1: Find the 12th term: 6, 10, 14, 18, ...

Arithmetic: \( d = 4 \), \( a_1 = 6 \)

\( a_{12} = 6 + (12-1)(4) = 6 + 44 = 50 \)

Problem 2: Find the 8th term: 1, 3, 9, 27, ...

Geometric: \( r = 3 \), \( a_1 = 1 \)

\( a_8 = 1 \cdot 3^{8-1} = 3^7 = 2187 \)

Problem 3: Write an expression for: 2, 7, 12, 17, ...

\( d = 5 \), \( a_1 = 2 \)

\( a_n = 2 + (n-1)(5) = 5n - 3 \)

Quick Reference: Sequences

Arithmetic Sequence:

Common Difference: \( d = a_2 - a_1 \)

Nth Term: \( a_n = a_1 + (n-1)d \) or \( a_n = dn + c \)

Example: 3, 7, 11, 15, 19 (d = 4)

Geometric Sequence:

Common Ratio: \( r = \frac{a_2}{a_1} \)

Nth Term: \( a_n = a_1 \cdot r^{n-1} \)

Example: 2, 6, 18, 54, 162 (r = 3)

Quick Identification:

• Arithmetic: Constant difference between terms
• Geometric: Constant ratio between terms

💡 Key Tips for Sequences

• ✓ Arithmetic: add/subtract same number each time
• ✓ Geometric: multiply/divide by same number each time
• ✓ Find d: subtract consecutive terms
• ✓ Find r: divide consecutive terms
• ✓ Arithmetic formula: a_n = a_1 + (n-1)d
• ✓ Geometric formula: a_n = a_1 · r^(n-1)
• ✓ n represents the term position (1st, 2nd, 3rd, etc.)
• ✓ Negative d → decreasing arithmetic sequence
• ✓ r between 0 and 1 → decreasing geometric sequence
• ✓ Negative r → alternating signs in geometric
• ✓ Always identify first term (a_1) before using formula
• ✓ Check your answer by calculating forward