🎯 Sequences and Patterns Problem Types
📊 Problem Pattern Catalog
Type 1: Find the nth Term
Pattern: Given a sequence, find the formula for the nth term Key Concept: Identify if it’s arithmetic, geometric, or other
Worked Example:
Find the 10th term of the sequence: 3, 7, 11, 15, 19, …
Solution: This is an arithmetic sequence with $a_1 = 3$ and $d = 4$.
$a_n = a_1 + (n-1)d = 3 + (n-1) \cdot 4 = 3 + 4n - 4 = 4n - 1$
$a_{10} = 4 \cdot 10 - 1 = 39$
Type 2: Find the Sum of Terms
Pattern: Calculate the sum of a certain number of terms Key Formula: Use appropriate sum formula for sequence type
Worked Example:
Find the sum of the first 8 terms of: 2, 6, 18, 54, …
Solution: This is a geometric sequence with $a_1 = 2$ and $r = 3$.
$S_n = \frac{a_1(1-r^n)}{1-r} = \frac{2(1-3^8)}{1-3} = \frac{2(1-6561)}{-2} = \frac{2(-6560)}{-2} = 6560$
Type 3: Pattern Recognition
Pattern: Find the rule for a given sequence Key Concept: Look for differences, ratios, or other patterns
Worked Example:
Find the next three terms: 1, 4, 9, 16, 25, …
Solution: Looking at the sequence: 1, 4, 9, 16, 25, …
These are perfect squares: $1^2, 2^2, 3^2, 4^2, 5^2, …$
So the next three terms are $6^2 = 36$, $7^2 = 49$, $8^2 = 64$
Type 4: Word Problems with Sequences
Pattern: Real-world problems involving sequences Key Concept: Translate the problem into sequence terms
Worked Example:
A ball is dropped from 100 feet. Each bounce reaches 3/4 of the previous height. How high does it reach on the 5th bounce?
Solution: This is a geometric sequence with $a_1 = 100$ and $r = \frac{3}{4}$.
$a_n = a_1 \cdot r^{n-1} = 100 \cdot \left(\frac{3}{4}\right)^{n-1}$
$a_5 = 100 \cdot \left(\frac{3}{4}\right)^4 = 100 \cdot \frac{81}{256} = \frac{8100}{256} = \frac{2025}{64} \approx 31.64$ feet
Type 5: Special Sequences
Pattern: Work with Fibonacci, triangular, or other special sequences Key Concept: Use specific formulas for special sequences
Worked Example:
Find the 8th triangular number.
Solution: Triangular numbers: $T_n = \frac{n(n+1)}{2}$
$T_8 = \frac{8 \cdot 9}{2} = \frac{72}{2} = 36$
🔍 Problem-Solving Strategy
- Identify the sequence type (arithmetic, geometric, special)
- Find the pattern by examining differences or ratios
- Write the general formula for the nth term
- Use appropriate formulas for sums or specific terms
- Check your work by verifying a few terms
⚠️ Common Mistakes
- Wrong sequence type identification
- Off-by-one errors in term numbers
- Sign errors in calculations
- Using wrong formula for sums
- Not recognizing special sequence patterns