Study Notes
Sequences in algebra involve finding patterns and rules to determine terms in a series. There are different types of sequences, including arithmetic, geometric, quadratic, and cubic sequences, each with its own method of finding terms and differences.
- Geometric Sequence — A sequence where each term is obtained by multiplying the previous term by a constant called the common ratio. Example: In the sequence 2, 6, 18, 54,..., each term is multiplied by 3.
- Arithmetic Sequence — A sequence where each term is obtained by adding a constant difference to the previous term. Example: In the sequence 6, 10, 14, 18,..., each term increases by 4.
- Quadratic Sequence — A sequence where the second difference between terms is constant, and the nth term is expressed as a quadratic expression. Example: In the sequence 2, 7, 14, 23, 34,..., the second difference is 2.
- Cubic Sequence — A sequence where the third difference between terms is constant, and the nth term is expressed as a cubic expression. Example: In the sequence 4, 16, 44, 94, 172,..., the third difference is 6.
Exam Tips
Key Definitions to Remember
- Geometric Sequence: Each term is found by multiplying the previous term by a constant ratio.
- Arithmetic Sequence: Each term is found by adding a constant difference to the previous term.
- Quadratic Sequence: A sequence with a constant second difference.
- Cubic Sequence: A sequence with a constant third difference.
Common Confusions
- Confusing the common ratio in geometric sequences with the common difference in arithmetic sequences.
- Misidentifying the type of sequence based on the differences between terms.
Typical Exam Questions
- What is the 1,000th term in the sequence 3, 7, 11,...? Use the formula for the nth term of an arithmetic sequence.
- What is the 10th term in the sequence where the nth term is 3n−10? Substitute n=10 into the formula.
- How many terms of the geometric sequence 2, 8, 32, 128,... are required to give a sum of 174,762? Use the formula for the sum of a geometric series.
What Examiners Usually Test
- Ability to identify and apply the correct formula for different types of sequences.
- Calculating specific terms in a sequence using the general term formula.
- Understanding and applying the concept of differences in sequences to find patterns.