📖Definition
A sequence is an ordered list of numbers following a specific pattern or rule. Each number is called a term. In an arithmetic sequence, the difference between consecutive terms is constant. In a geometric sequence, the ratio between consecutive terms is constant.
📐Formula
For arithmetic sequences, d is the common difference. For geometric sequences, r is the common ratio. aₙ is the nth term and a₁ is the first term.
📝Step-by-Step Guide
Identify the Pattern
Look for addition/subtraction (arithmetic) or multiplication/division (geometric) between terms.
Find the Common Difference or Ratio
Arithmetic: d = any term minus the previous term. Geometric: r = any term divided by the previous term.
Write the Formula
Use the appropriate formula for the sequence type.
Find Any Term
Substitute the term number (n) into the formula to find that term.
⚠️Common Mistakes to Avoid
- Confusing arithmetic and geometric sequences
- Using the wrong formula
- Off-by-one errors with term numbers
- Miscalculating the common difference or ratio
- Forgetting n-1 in the formulas
✏️Practice Problems
Find the next term: 2, 5, 8, 11, ...
Answer: 14 (arithmetic, d = 3)
Find the 10th term of: 3, 6, 12, 24, ...
Answer: 1536 (geometric, r = 2)
In an arithmetic sequence, the 3rd term is 11 and the 7th term is 27. Find the first term.
Answer: a₁ = 3
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