Sequences - Ninth Grade Math
Introduction to Sequences
Sequence: An ordered list of numbers following a specific pattern
Term: Each number in a sequence
Notation: $a_1, a_2, a_3, ..., a_n$ where $n$ represents the position (index)
$a_1$: First term
$a_n$: nth term (general term)
Term: Each number in a sequence
Notation: $a_1, a_2, a_3, ..., a_n$ where $n$ represents the position (index)
$a_1$: First term
$a_n$: nth term (general term)
Types of Sequences:
• Arithmetic Sequence: Constant difference between consecutive terms
• Geometric Sequence: Constant ratio between consecutive terms
• Other: Fibonacci, quadratic, etc.
• Arithmetic Sequence: Constant difference between consecutive terms
• Geometric Sequence: Constant ratio between consecutive terms
• Other: Fibonacci, quadratic, etc.
1. Identify Arithmetic and Geometric Sequences
Arithmetic Sequence: Add (or subtract) the same number to get the next term
Common Difference (d): The constant added each time
Geometric Sequence: Multiply by the same number to get the next term
Common Ratio (r): The constant multiplied each time
Common Difference (d): The constant added each time
Geometric Sequence: Multiply by the same number to get the next term
Common Ratio (r): The constant multiplied each time
How to Identify:
Arithmetic Sequence:
$$a_2 - a_1 = a_3 - a_2 = a_4 - a_3 = d$$
(Differences are equal)
Geometric Sequence:
$$\frac{a_2}{a_1} = \frac{a_3}{a_2} = \frac{a_4}{a_3} = r$$
(Ratios are equal)
Arithmetic Sequence:
$$a_2 - a_1 = a_3 - a_2 = a_4 - a_3 = d$$
(Differences are equal)
Geometric Sequence:
$$\frac{a_2}{a_1} = \frac{a_3}{a_2} = \frac{a_4}{a_3} = r$$
(Ratios are equal)
Example 1: Identify the sequence: 3, 7, 11, 15, 19, ...
Check differences:
$7 - 3 = 4$
$11 - 7 = 4$
$15 - 11 = 4$
$19 - 15 = 4$
Answer: ARITHMETIC (common difference $d = 4$)
Check differences:
$7 - 3 = 4$
$11 - 7 = 4$
$15 - 11 = 4$
$19 - 15 = 4$
Answer: ARITHMETIC (common difference $d = 4$)
Example 2: Identify: 2, 6, 18, 54, 162, ...
Check differences: $4, 12, 36, 108$ (not constant)
Check ratios:
$\frac{6}{2} = 3$
$\frac{18}{6} = 3$
$\frac{54}{18} = 3$
$\frac{162}{54} = 3$
Answer: GEOMETRIC (common ratio $r = 3$)
Check differences: $4, 12, 36, 108$ (not constant)
Check ratios:
$\frac{6}{2} = 3$
$\frac{18}{6} = 3$
$\frac{54}{18} = 3$
$\frac{162}{54} = 3$
Answer: GEOMETRIC (common ratio $r = 3$)
Example 3: Identify: 5, 10, 20, 40, 80, ...
Ratios: $\frac{10}{5} = 2$, $\frac{20}{10} = 2$, $\frac{40}{20} = 2$
Answer: GEOMETRIC ($r = 2$)
Ratios: $\frac{10}{5} = 2$, $\frac{20}{10} = 2$, $\frac{40}{20} = 2$
Answer: GEOMETRIC ($r = 2$)
Example 4: Identify: 100, 85, 70, 55, 40, ...
Differences: $-15, -15, -15, -15$
Answer: ARITHMETIC ($d = -15$)
Differences: $-15, -15, -15, -15$
Answer: ARITHMETIC ($d = -15$)
2. Arithmetic Sequences
Arithmetic Sequence: A sequence where the difference between consecutive terms is constant
Common Difference (d): $d = a_{n+1} - a_n$
Common Difference (d): $d = a_{n+1} - a_n$
Explicit Formula (nth Term):
$$a_n = a_1 + (n - 1)d$$
where:
• $a_n$ = nth term
• $a_1$ = first term
• $n$ = position number
• $d$ = common difference
Common Difference Formula:
$$d = a_2 - a_1$$
$$a_n = a_1 + (n - 1)d$$
where:
• $a_n$ = nth term
• $a_1$ = first term
• $n$ = position number
• $d$ = common difference
Common Difference Formula:
$$d = a_2 - a_1$$
Example 1: Find the 15th term of: 5, 9, 13, 17, ...
Identify values:
$a_1 = 5$
$d = 9 - 5 = 4$
$n = 15$
Use formula:
$a_{15} = 5 + (15 - 1)(4)$
$a_{15} = 5 + (14)(4)$
$a_{15} = 5 + 56 = 61$
Answer: $a_{15} = 61$
Identify values:
$a_1 = 5$
$d = 9 - 5 = 4$
$n = 15$
Use formula:
$a_{15} = 5 + (15 - 1)(4)$
$a_{15} = 5 + (14)(4)$
$a_{15} = 5 + 56 = 61$
Answer: $a_{15} = 61$
Example 2: Find the 20th term of: 100, 93, 86, 79, ...
$a_1 = 100$, $d = -7$, $n = 20$
$a_{20} = 100 + (20-1)(-7) = 100 + (19)(-7) = 100 - 133 = -33$
Answer: $-33$
$a_1 = 100$, $d = -7$, $n = 20$
$a_{20} = 100 + (20-1)(-7) = 100 + (19)(-7) = 100 - 133 = -33$
Answer: $-33$
Example 3: Find which term equals 47 in: 3, 7, 11, 15, ...
$a_1 = 3$, $d = 4$, $a_n = 47$
$47 = 3 + (n-1)(4)$
$47 = 3 + 4n - 4$
$47 = 4n - 1$
$48 = 4n$
$n = 12$
Answer: 12th term
$a_1 = 3$, $d = 4$, $a_n = 47$
$47 = 3 + (n-1)(4)$
$47 = 3 + 4n - 4$
$47 = 4n - 1$
$48 = 4n$
$n = 12$
Answer: 12th term
3. Geometric Sequences
Geometric Sequence: A sequence where the ratio between consecutive terms is constant
Common Ratio (r): $r = \frac{a_{n+1}}{a_n}$
Common Ratio (r): $r = \frac{a_{n+1}}{a_n}$
Explicit Formula (nth Term):
$$a_n = a_1 \cdot r^{n-1}$$
where:
• $a_n$ = nth term
• $a_1$ = first term
• $n$ = position number
• $r$ = common ratio
Common Ratio Formula:
$$r = \frac{a_2}{a_1}$$
$$a_n = a_1 \cdot r^{n-1}$$
where:
• $a_n$ = nth term
• $a_1$ = first term
• $n$ = position number
• $r$ = common ratio
Common Ratio Formula:
$$r = \frac{a_2}{a_1}$$
Example 1: Find the 7th term of: 3, 6, 12, 24, ...
Identify values:
$a_1 = 3$
$r = \frac{6}{3} = 2$
$n = 7$
Use formula:
$a_7 = 3 \cdot 2^{7-1}$
$a_7 = 3 \cdot 2^6$
$a_7 = 3 \cdot 64 = 192$
Answer: $a_7 = 192$
Identify values:
$a_1 = 3$
$r = \frac{6}{3} = 2$
$n = 7$
Use formula:
$a_7 = 3 \cdot 2^{7-1}$
$a_7 = 3 \cdot 2^6$
$a_7 = 3 \cdot 64 = 192$
Answer: $a_7 = 192$
Example 2: Find the 6th term of: 5, 10, 20, 40, ...
$a_1 = 5$, $r = 2$, $n = 6$
$a_6 = 5 \cdot 2^{6-1} = 5 \cdot 2^5 = 5 \cdot 32 = 160$
Answer: $160$
$a_1 = 5$, $r = 2$, $n = 6$
$a_6 = 5 \cdot 2^{6-1} = 5 \cdot 2^5 = 5 \cdot 32 = 160$
Answer: $160$
Example 3: Find the 5th term of: 100, 50, 25, 12.5, ...
$a_1 = 100$, $r = \frac{50}{100} = \frac{1}{2} = 0.5$
$a_5 = 100 \cdot (0.5)^{5-1} = 100 \cdot (0.5)^4 = 100 \cdot 0.0625 = 6.25$
Answer: $6.25$
$a_1 = 100$, $r = \frac{50}{100} = \frac{1}{2} = 0.5$
$a_5 = 100 \cdot (0.5)^{5-1} = 100 \cdot (0.5)^4 = 100 \cdot 0.0625 = 6.25$
Answer: $6.25$
Example 4: Find the 4th term of: $2, -6, 18, -54, ...$
$a_1 = 2$, $r = \frac{-6}{2} = -3$
$a_4 = 2 \cdot (-3)^{4-1} = 2 \cdot (-3)^3 = 2 \cdot (-27) = -54$ ✓ (verify)
$a_5 = 2 \cdot (-3)^4 = 2 \cdot 81 = 162$
Answer: 5th term is $162$
$a_1 = 2$, $r = \frac{-6}{2} = -3$
$a_4 = 2 \cdot (-3)^{4-1} = 2 \cdot (-3)^3 = 2 \cdot (-27) = -54$ ✓ (verify)
$a_5 = 2 \cdot (-3)^4 = 2 \cdot 81 = 162$
Answer: 5th term is $162$
4. Evaluate Variable Expressions for Sequences
Variable Expression: A formula with variables that represents the nth term
Example 1: Given $a_n = 3n + 2$, find $a_5$
Substitute $n = 5$:
$a_5 = 3(5) + 2 = 15 + 2 = 17$
Answer: $a_5 = 17$
Substitute $n = 5$:
$a_5 = 3(5) + 2 = 15 + 2 = 17$
Answer: $a_5 = 17$
Example 2: Given $a_n = 2^n$, find the first 4 terms
$a_1 = 2^1 = 2$
$a_2 = 2^2 = 4$
$a_3 = 2^3 = 8$
$a_4 = 2^4 = 16$
Sequence: 2, 4, 8, 16, ...
$a_1 = 2^1 = 2$
$a_2 = 2^2 = 4$
$a_3 = 2^3 = 8$
$a_4 = 2^4 = 16$
Sequence: 2, 4, 8, 16, ...
Example 3: Given $a_n = n^2 - 1$, find $a_6$
$a_6 = 6^2 - 1 = 36 - 1 = 35$
Answer: $35$
$a_6 = 6^2 - 1 = 36 - 1 = 35$
Answer: $35$
Example 4: Given $a_n = \frac{n}{n+1}$, find first 3 terms
$a_1 = \frac{1}{1+1} = \frac{1}{2}$
$a_2 = \frac{2}{2+1} = \frac{2}{3}$
$a_3 = \frac{3}{3+1} = \frac{3}{4}$
Sequence: $\frac{1}{2}, \frac{2}{3}, \frac{3}{4}, ...$
$a_1 = \frac{1}{1+1} = \frac{1}{2}$
$a_2 = \frac{2}{2+1} = \frac{2}{3}$
$a_3 = \frac{3}{3+1} = \frac{3}{4}$
Sequence: $\frac{1}{2}, \frac{2}{3}, \frac{3}{4}, ...$
5. Evaluate Recursive Formulas for Sequences
Recursive Formula: A formula that defines each term using the previous term(s)
Format: $a_n = f(a_{n-1})$ or $a_n = f(a_{n-1}, a_{n-2})$
Required: Initial term(s) must be given
Format: $a_n = f(a_{n-1})$ or $a_n = f(a_{n-1}, a_{n-2})$
Required: Initial term(s) must be given
Common Recursive Forms:
Arithmetic (Recursive):
$$a_1 = \text{(first term)}$$ $$a_n = a_{n-1} + d$$
Geometric (Recursive):
$$a_1 = \text{(first term)}$$ $$a_n = a_{n-1} \cdot r$$
Arithmetic (Recursive):
$$a_1 = \text{(first term)}$$ $$a_n = a_{n-1} + d$$
Geometric (Recursive):
$$a_1 = \text{(first term)}$$ $$a_n = a_{n-1} \cdot r$$
Example 1: Given $a_1 = 5$ and $a_n = a_{n-1} + 3$, find first 5 terms
$a_1 = 5$
$a_2 = a_1 + 3 = 5 + 3 = 8$
$a_3 = a_2 + 3 = 8 + 3 = 11$
$a_4 = a_3 + 3 = 11 + 3 = 14$
$a_5 = a_4 + 3 = 14 + 3 = 17$
Sequence: 5, 8, 11, 14, 17, ...
$a_1 = 5$
$a_2 = a_1 + 3 = 5 + 3 = 8$
$a_3 = a_2 + 3 = 8 + 3 = 11$
$a_4 = a_3 + 3 = 11 + 3 = 14$
$a_5 = a_4 + 3 = 14 + 3 = 17$
Sequence: 5, 8, 11, 14, 17, ...
Example 2: Given $a_1 = 2$ and $a_n = 2 \cdot a_{n-1}$, find first 5 terms
$a_1 = 2$
$a_2 = 2 \cdot a_1 = 2 \cdot 2 = 4$
$a_3 = 2 \cdot a_2 = 2 \cdot 4 = 8$
$a_4 = 2 \cdot a_3 = 2 \cdot 8 = 16$
$a_5 = 2 \cdot a_4 = 2 \cdot 16 = 32$
Sequence: 2, 4, 8, 16, 32, ...
$a_1 = 2$
$a_2 = 2 \cdot a_1 = 2 \cdot 2 = 4$
$a_3 = 2 \cdot a_2 = 2 \cdot 4 = 8$
$a_4 = 2 \cdot a_3 = 2 \cdot 8 = 16$
$a_5 = 2 \cdot a_4 = 2 \cdot 16 = 32$
Sequence: 2, 4, 8, 16, 32, ...
Example 3: Fibonacci sequence: $a_1 = 1$, $a_2 = 1$, $a_n = a_{n-1} + a_{n-2}$
$a_1 = 1$
$a_2 = 1$
$a_3 = a_2 + a_1 = 1 + 1 = 2$
$a_4 = a_3 + a_2 = 2 + 1 = 3$
$a_5 = a_4 + a_3 = 3 + 2 = 5$
$a_6 = a_5 + a_4 = 5 + 3 = 8$
Sequence: 1, 1, 2, 3, 5, 8, ...
$a_1 = 1$
$a_2 = 1$
$a_3 = a_2 + a_1 = 1 + 1 = 2$
$a_4 = a_3 + a_2 = 2 + 1 = 3$
$a_5 = a_4 + a_3 = 3 + 2 = 5$
$a_6 = a_5 + a_4 = 5 + 3 = 8$
Sequence: 1, 1, 2, 3, 5, 8, ...
6. Identify a Sequence as Explicit or Recursive
Explicit Formula: Gives the nth term directly using only $n$
• Can find any term without knowing previous terms
• Example: $a_n = 3n + 5$
Recursive Formula: Defines term using previous term(s)
• Must know previous terms to find next term
• Example: $a_n = a_{n-1} + 3$, with $a_1 = 8$
• Can find any term without knowing previous terms
• Example: $a_n = 3n + 5$
Recursive Formula: Defines term using previous term(s)
• Must know previous terms to find next term
• Example: $a_n = a_{n-1} + 3$, with $a_1 = 8$
How to Identify:
Explicit:
• Formula contains only $n$ (and constants)
• No reference to previous terms like $a_{n-1}$
• Can directly calculate any term
Recursive:
• Formula contains $a_{n-1}$, $a_{n-2}$, etc.
• Requires initial term(s)
• Must calculate terms in order
Explicit:
• Formula contains only $n$ (and constants)
• No reference to previous terms like $a_{n-1}$
• Can directly calculate any term
Recursive:
• Formula contains $a_{n-1}$, $a_{n-2}$, etc.
• Requires initial term(s)
• Must calculate terms in order
Example 1: Classify: $a_n = 5n - 2$
Contains only $n$ (no previous terms)
Answer: EXPLICIT
Contains only $n$ (no previous terms)
Answer: EXPLICIT
Example 2: Classify: $a_1 = 10$, $a_n = a_{n-1} - 4$
Uses previous term $a_{n-1}$
Answer: RECURSIVE
Uses previous term $a_{n-1}$
Answer: RECURSIVE
Example 3: Classify: $a_n = 2 \cdot 3^{n-1}$
Direct formula with $n$
Answer: EXPLICIT
Direct formula with $n$
Answer: EXPLICIT
Example 4: Classify: $a_1 = 7$, $a_n = 2a_{n-1}$
Depends on previous term
Answer: RECURSIVE
Depends on previous term
Answer: RECURSIVE
7. Write Variable Expressions for Arithmetic Sequences
Steps to Write Arithmetic Explicit Formula:
Step 1: Find first term $a_1$
Step 2: Find common difference $d = a_2 - a_1$
Step 3: Substitute into formula: $a_n = a_1 + (n-1)d$
Step 4: Simplify if possible
Step 1: Find first term $a_1$
Step 2: Find common difference $d = a_2 - a_1$
Step 3: Substitute into formula: $a_n = a_1 + (n-1)d$
Step 4: Simplify if possible
Example 1: Write formula for: 4, 9, 14, 19, 24, ...
$a_1 = 4$
$d = 9 - 4 = 5$
$a_n = 4 + (n-1)(5)$
$a_n = 4 + 5n - 5$
$a_n = 5n - 1$
Answer: $a_n = 5n - 1$
$a_1 = 4$
$d = 9 - 4 = 5$
$a_n = 4 + (n-1)(5)$
$a_n = 4 + 5n - 5$
$a_n = 5n - 1$
Answer: $a_n = 5n - 1$
Example 2: Write formula for: 20, 16, 12, 8, 4, ...
$a_1 = 20$, $d = -4$
$a_n = 20 + (n-1)(-4)$
$a_n = 20 - 4n + 4$
$a_n = 24 - 4n$
Answer: $a_n = 24 - 4n$
$a_1 = 20$, $d = -4$
$a_n = 20 + (n-1)(-4)$
$a_n = 20 - 4n + 4$
$a_n = 24 - 4n$
Answer: $a_n = 24 - 4n$
8. Write Variable Expressions for Geometric Sequences
Steps to Write Geometric Explicit Formula:
Step 1: Find first term $a_1$
Step 2: Find common ratio $r = \frac{a_2}{a_1}$
Step 3: Substitute into formula: $a_n = a_1 \cdot r^{n-1}$
Step 4: Simplify if needed
Step 1: Find first term $a_1$
Step 2: Find common ratio $r = \frac{a_2}{a_1}$
Step 3: Substitute into formula: $a_n = a_1 \cdot r^{n-1}$
Step 4: Simplify if needed
Example 1: Write formula for: 3, 12, 48, 192, ...
$a_1 = 3$
$r = \frac{12}{3} = 4$
$a_n = 3 \cdot 4^{n-1}$
Answer: $a_n = 3 \cdot 4^{n-1}$
$a_1 = 3$
$r = \frac{12}{3} = 4$
$a_n = 3 \cdot 4^{n-1}$
Answer: $a_n = 3 \cdot 4^{n-1}$
Example 2: Write formula for: 5, 15, 45, 135, ...
$a_1 = 5$, $r = 3$
$a_n = 5 \cdot 3^{n-1}$
Answer: $a_n = 5 \cdot 3^{n-1}$
$a_1 = 5$, $r = 3$
$a_n = 5 \cdot 3^{n-1}$
Answer: $a_n = 5 \cdot 3^{n-1}$
Example 3: Write formula for: 100, 50, 25, 12.5, ...
$a_1 = 100$, $r = \frac{1}{2}$
$a_n = 100 \cdot \left(\frac{1}{2}\right)^{n-1}$
Answer: $a_n = 100 \cdot \left(\frac{1}{2}\right)^{n-1}$
$a_1 = 100$, $r = \frac{1}{2}$
$a_n = 100 \cdot \left(\frac{1}{2}\right)^{n-1}$
Answer: $a_n = 100 \cdot \left(\frac{1}{2}\right)^{n-1}$
9. Write a Formula for a Recursive Sequence
Steps to Write Recursive Formula:
Step 1: Identify the pattern (how to get from one term to next)
Step 2: Write initial term(s): $a_1 = ...$
Step 3: Write relationship: $a_n = f(a_{n-1})$
Step 4: Verify with given terms
Step 1: Identify the pattern (how to get from one term to next)
Step 2: Write initial term(s): $a_1 = ...$
Step 3: Write relationship: $a_n = f(a_{n-1})$
Step 4: Verify with given terms
Example 1: Write recursive formula for: 7, 11, 15, 19, 23, ...
Pattern: Add 4 each time
$a_1 = 7$
$a_n = a_{n-1} + 4$
Answer: $a_1 = 7$, $a_n = a_{n-1} + 4$
Pattern: Add 4 each time
$a_1 = 7$
$a_n = a_{n-1} + 4$
Answer: $a_1 = 7$, $a_n = a_{n-1} + 4$
Example 2: Write recursive formula for: 2, 10, 50, 250, ...
Pattern: Multiply by 5 each time
$a_1 = 2$
$a_n = 5 \cdot a_{n-1}$
Answer: $a_1 = 2$, $a_n = 5a_{n-1}$
Pattern: Multiply by 5 each time
$a_1 = 2$
$a_n = 5 \cdot a_{n-1}$
Answer: $a_1 = 2$, $a_n = 5a_{n-1}$
Example 3: Write recursive formula for: 1, 4, 9, 16, 25, ...
Pattern: Differences are 3, 5, 7, 9 (add 2 more each time)
This is not simple arithmetic or geometric
$a_1 = 1$
$a_n = a_{n-1} + (2n - 1)$
Or recognize: $a_n = n^2$ (explicit is easier!)
Pattern: Differences are 3, 5, 7, 9 (add 2 more each time)
This is not simple arithmetic or geometric
$a_1 = 1$
$a_n = a_{n-1} + (2n - 1)$
Or recognize: $a_n = n^2$ (explicit is easier!)
10. Convert Recursive to Explicit Formula
Goal: Transform a recursive formula into an explicit formula
For Arithmetic Sequences:
If $a_1 = c$ and $a_n = a_{n-1} + d$
Then explicit formula is: $a_n = c + (n-1)d$
For Geometric Sequences:
If $a_1 = c$ and $a_n = r \cdot a_{n-1}$
Then explicit formula is: $a_n = c \cdot r^{n-1}$
If $a_1 = c$ and $a_n = a_{n-1} + d$
Then explicit formula is: $a_n = c + (n-1)d$
For Geometric Sequences:
If $a_1 = c$ and $a_n = r \cdot a_{n-1}$
Then explicit formula is: $a_n = c \cdot r^{n-1}$
Example 1: Convert to explicit: $a_1 = 6$, $a_n = a_{n-1} + 4$
Identify: Arithmetic with $a_1 = 6$, $d = 4$
$a_n = 6 + (n-1)(4)$
$a_n = 6 + 4n - 4$
$a_n = 4n + 2$
Answer: $a_n = 4n + 2$
Identify: Arithmetic with $a_1 = 6$, $d = 4$
$a_n = 6 + (n-1)(4)$
$a_n = 6 + 4n - 4$
$a_n = 4n + 2$
Answer: $a_n = 4n + 2$
Example 2: Convert to explicit: $a_1 = 5$, $a_n = 3a_{n-1}$
Identify: Geometric with $a_1 = 5$, $r = 3$
$a_n = 5 \cdot 3^{n-1}$
Answer: $a_n = 5 \cdot 3^{n-1}$
Identify: Geometric with $a_1 = 5$, $r = 3$
$a_n = 5 \cdot 3^{n-1}$
Answer: $a_n = 5 \cdot 3^{n-1}$
Example 3: Convert: $a_1 = 100$, $a_n = a_{n-1} - 7$
Arithmetic: $a_1 = 100$, $d = -7$
$a_n = 100 + (n-1)(-7)$
$a_n = 100 - 7n + 7$
$a_n = 107 - 7n$
Answer: $a_n = 107 - 7n$
Arithmetic: $a_1 = 100$, $d = -7$
$a_n = 100 + (n-1)(-7)$
$a_n = 100 - 7n + 7$
$a_n = 107 - 7n$
Answer: $a_n = 107 - 7n$
11. Convert Explicit to Recursive Formula
For Arithmetic (form $a_n = mn + b$):
Find $a_1$ by substituting $n = 1$
Common difference $d = m$ (coefficient of $n$)
Recursive: $a_1 = ...$, $a_n = a_{n-1} + d$
For Geometric (form $a_n = c \cdot r^{n-1}$):
First term is $a_1 = c$
Common ratio is $r$
Recursive: $a_1 = c$, $a_n = r \cdot a_{n-1}$
Find $a_1$ by substituting $n = 1$
Common difference $d = m$ (coefficient of $n$)
Recursive: $a_1 = ...$, $a_n = a_{n-1} + d$
For Geometric (form $a_n = c \cdot r^{n-1}$):
First term is $a_1 = c$
Common ratio is $r$
Recursive: $a_1 = c$, $a_n = r \cdot a_{n-1}$
Example 1: Convert to recursive: $a_n = 3n + 5$
Find $a_1$: $a_1 = 3(1) + 5 = 8$
Common difference: $d = 3$
Answer: $a_1 = 8$, $a_n = a_{n-1} + 3$
Find $a_1$: $a_1 = 3(1) + 5 = 8$
Common difference: $d = 3$
Answer: $a_1 = 8$, $a_n = a_{n-1} + 3$
Example 2: Convert to recursive: $a_n = 2 \cdot 5^{n-1}$
First term: $a_1 = 2 \cdot 5^{1-1} = 2 \cdot 5^0 = 2$
Common ratio: $r = 5$
Answer: $a_1 = 2$, $a_n = 5a_{n-1}$
First term: $a_1 = 2 \cdot 5^{1-1} = 2 \cdot 5^0 = 2$
Common ratio: $r = 5$
Answer: $a_1 = 2$, $a_n = 5a_{n-1}$
Example 3: Convert to recursive: $a_n = 50 - 6n$
$a_1 = 50 - 6(1) = 44$
$d = -6$
Answer: $a_1 = 44$, $a_n = a_{n-1} - 6$
$a_1 = 50 - 6(1) = 44$
$d = -6$
Answer: $a_1 = 44$, $a_n = a_{n-1} - 6$
Summary: Formulas Comparison
| Sequence Type | Explicit Formula | Recursive Formula | How to Identify |
|---|---|---|---|
| Arithmetic | $a_n = a_1 + (n-1)d$ | $a_1 = ...$ $a_n = a_{n-1} + d$ | Constant difference |
| Geometric | $a_n = a_1 \cdot r^{n-1}$ | $a_1 = ...$ $a_n = r \cdot a_{n-1}$ | Constant ratio |
Formula Comparison Chart
| Feature | Explicit Formula | Recursive Formula |
|---|---|---|
| Definition | Direct formula using position $n$ | Formula using previous term(s) |
| Contains | Only $n$ and constants | $a_{n-1}$, $a_{n-2}$, etc. |
| Advantage | Find any term directly | Shows relationship between terms |
| Disadvantage | May be hard to find pattern | Must calculate all previous terms |
| Example (Arithmetic) | $a_n = 5n - 2$ | $a_1 = 3$, $a_n = a_{n-1} + 5$ |
| Example (Geometric) | $a_n = 2 \cdot 3^{n-1}$ | $a_1 = 2$, $a_n = 3a_{n-1}$ |
Mixed Review Examples
Problem 1: Identify the sequence and find the 10th term: 5, 8, 11, 14, 17, ...
Type: Arithmetic ($d = 3$)
Formula: $a_n = 5 + (n-1)(3) = 3n + 2$
10th term: $a_{10} = 3(10) + 2 = 32$
Type: Arithmetic ($d = 3$)
Formula: $a_n = 5 + (n-1)(3) = 3n + 2$
10th term: $a_{10} = 3(10) + 2 = 32$
Problem 2: Write both explicit and recursive formulas: 2, 6, 18, 54, ...
Type: Geometric ($r = 3$)
Explicit: $a_n = 2 \cdot 3^{n-1}$
Recursive: $a_1 = 2$, $a_n = 3a_{n-1}$
Type: Geometric ($r = 3$)
Explicit: $a_n = 2 \cdot 3^{n-1}$
Recursive: $a_1 = 2$, $a_n = 3a_{n-1}$
Problem 3: Given $a_n = 7n - 3$, convert to recursive form
$a_1 = 7(1) - 3 = 4$
$d = 7$
Recursive: $a_1 = 4$, $a_n = a_{n-1} + 7$
$a_1 = 7(1) - 3 = 4$
$d = 7$
Recursive: $a_1 = 4$, $a_n = a_{n-1} + 7$
Problem 4: Given $a_1 = 10$, $a_n = a_{n-1} - 2$, find explicit formula
Arithmetic: $a_1 = 10$, $d = -2$
$a_n = 10 + (n-1)(-2)$
$a_n = 10 - 2n + 2$
Explicit: $a_n = 12 - 2n$
Arithmetic: $a_1 = 10$, $d = -2$
$a_n = 10 + (n-1)(-2)$
$a_n = 10 - 2n + 2$
Explicit: $a_n = 12 - 2n$
Key Concepts to Remember
Arithmetic Sequences:
• Add/subtract same amount each time
• Formula: $a_n = a_1 + (n-1)d$
• Example: 3, 7, 11, 15, ... ($d = 4$)
Geometric Sequences:
• Multiply by same amount each time
• Formula: $a_n = a_1 \cdot r^{n-1}$
• Example: 3, 6, 12, 24, ... ($r = 2$)
Explicit vs Recursive:
• Explicit: Direct formula with $n$
• Recursive: Uses previous term(s)
• Can convert between the two!
• Add/subtract same amount each time
• Formula: $a_n = a_1 + (n-1)d$
• Example: 3, 7, 11, 15, ... ($d = 4$)
Geometric Sequences:
• Multiply by same amount each time
• Formula: $a_n = a_1 \cdot r^{n-1}$
• Example: 3, 6, 12, 24, ... ($r = 2$)
Explicit vs Recursive:
• Explicit: Direct formula with $n$
• Recursive: Uses previous term(s)
• Can convert between the two!
Success Tips for Sequences:
✓ Always check if sequence is arithmetic or geometric first
✓ Find the pattern: constant difference or constant ratio?
✓ For arithmetic: $a_n = a_1 + (n-1)d$
✓ For geometric: $a_n = a_1 \cdot r^{n-1}$
✓ Explicit formulas use only $n$, recursive uses previous terms
✓ To convert explicit to recursive: find $a_1$ and the pattern
✓ To convert recursive to explicit: identify if arithmetic or geometric
✓ Check your formula by testing with known terms
✓ Remember: position number $n$ starts at 1
✓ Practice identifying patterns in real-world contexts
✓ Always check if sequence is arithmetic or geometric first
✓ Find the pattern: constant difference or constant ratio?
✓ For arithmetic: $a_n = a_1 + (n-1)d$
✓ For geometric: $a_n = a_1 \cdot r^{n-1}$
✓ Explicit formulas use only $n$, recursive uses previous terms
✓ To convert explicit to recursive: find $a_1$ and the pattern
✓ To convert recursive to explicit: identify if arithmetic or geometric
✓ Check your formula by testing with known terms
✓ Remember: position number $n$ starts at 1
✓ Practice identifying patterns in real-world contexts