Sequences and Series
8.1 Introduction
Sequences and series are fundamental concepts in mathematics that deal with ordered lists of numbers and their sums.
Key Concepts:
- Sequence: An ordered list of numbers (terms)
- Series: The sum of terms of a sequence
- Applications in finance, physics, computer science, and more
8.2 Sequences
A sequence is a function whose domain is the set of natural numbers.
Types of Sequences:
| Type | Definition | Example |
|---|---|---|
| Arithmetic | an = a1 + (n-1)d | 2, 5, 8, 11,... (d=3) |
| Geometric | an = a1·rn-1 | 3, 6, 12, 24,... (r=2) |
| Fibonacci | an = an-1 + an-2 | 1, 1, 2, 3, 5, 8,... |
Example:
Find the 10th term of the arithmetic sequence: 7, 11, 15, 19,...
a1 = 7, d = 4
a10 = 7 + (10-1)×4 = 7 + 36 = 43
8.3 Series
The sum of terms of a sequence is called a series.
Notation: Sn = a1 + a2 + ... + an
Arithmetic Series Sum:
Sn = n/2 [2a1 + (n-1)d] = n/2 (a1 + an)
Visualization of Arithmetic Series:
S = 1 + 2 + 3 + ... + n = n(n+1)/2
[Triangular number diagram]
Example:
Find the sum of first 20 terms of: 5 + 9 + 13 + 17 + ...
a1 = 5, d = 4
S20 = 20/2 [2×5 + (20-1)×4] = 10 [10 + 76] = 860
8.4 Geometric Progression (G.P.)
A sequence where each term after the first is found by multiplying the previous term by a constant ratio (r).
General Term:
an = a1·rn-1
Finite Geometric Series Sum:
Sn = a1(1 - rn)/(1 - r) when r ≠ 1
Infinite Geometric Series Sum (|r| < 1):
S = a1/(1 - r)
Example 1:
Find the sum of first 5 terms of: 2 + 6 + 18 + 54 + ...
a1 = 2, r = 3
S5 = 2(1 - 35)/(1 - 3) = 2(1 - 243)/(-2) = 242
Example 2:
Find the sum to infinity: 1 + 1/2 + 1/4 + 1/8 + ...
a1 = 1, r = 1/2 (|r| < 1)
S = 1/(1 - 1/2) = 2
8.5 Relationship Between A.M. and G.M.
For any two positive numbers a and b:
(A.M.) = (a + b)/2
(G.M.) = √(ab)
Important Inequality:
A.M. ≥ G.M. ≥ H.M.
(Arithmetic Mean ≥ Geometric Mean ≥ Harmonic Mean)
Equality holds when a = b
Example:
For numbers 4 and 16:
A.M. = (4 + 16)/2 = 10
G.M. = √(4×16) = √64 = 8
10 ≥ 8 (A.M. ≥ G.M.)
Applications:
- Population growth modeling (G.P.)
- Loan amortization (A.P.)
- Optimization problems (A.M.-G.M. inequality)
- Fractal geometry (infinite series)
Practice Problem:
Insert 3 geometric means between 3 and 48.
Solution: Sequence is 3, 3r, 3r2, 3r3, 48
3r4 = 48 ⇒ r4 = 16 ⇒ r = 2
The means are 6, 12, 24