Understanding Sequences
A sequence is an ordered list of numbers that follows a specific pattern or rule. The two most common types of sequences are arithmetic sequences (constant difference between terms) and geometric sequences (constant ratio between terms).
Types of Sequences
Arithmetic Sequence
Each term differs from the previous by a constant amount (common difference d).
a_n = a_1 + (n-1)d
Geometric Sequence
Each term is multiplied by a constant factor (common ratio r).
a_n = a_1 × r^(n-1)
Arithmetic Sum
Sum of the first n terms of an arithmetic sequence.
S_n = n/2 × (a_1 + a_n)
Geometric Sum
Sum of the first n terms of a geometric sequence.
S_n = a_1(1 - r^n)/(1 - r)
How to Identify Sequence Type
- Calculate the differences between consecutive terms. If all differences are equal, the sequence is arithmetic.
- If not arithmetic, calculate the ratios between consecutive terms. If all ratios are equal, the sequence is geometric.
- If neither condition is met, the sequence may follow a more complex pattern (quadratic, etc.).
Examples
Arithmetic Sequence
2, 5, 8, 11, 14 has common difference d = 3. The nth term is a_n = 2 + (n-1)(3) = 3n - 1.
Geometric Sequence
3, 6, 12, 24, 48 has common ratio r = 2. The nth term is a_n = 3 × 2^(n-1).
Applications
- Finance: compound interest follows a geometric sequence.
- Physics: uniformly accelerated motion involves arithmetic sequences.
- Computer science: algorithm analysis uses sequences.
- Nature: population growth models use geometric sequences.