In mathematics, we often study patterns. A pattern of numbers arranged in a specific order is called a Sequence. If you decide to add all those numbers together, you create a Series.
This topic bridges the gap between simple arithmetic and calculus. It allows us to calculate loan payments, model population growth, and even determine how a bouncing ball eventually comes to rest.
1. What is a Sequence?
A sequence is simply an ordered list of numbers. Each number in the list is called a term. We usually denote terms as a1 (first term), a2 (second term), and so on.
Sequences can be finite (they end) or infinite (they go on forever).
2. Arithmetic Sequences
An arithmetic sequence changes by adding (or subtracting) the same value each time. This value is called the Common Difference (d).
Example: 5, 8, 11, 14...
The difference (d) is +3.
The General Formula: To find any term (an) without listing them all out:
If we want the 50th term of the example above:
a50 = 5 + (49)3 = 5 + 147 = 152.
3. Geometric Sequences
A geometric sequence changes by multiplying by the same value each time. This value is called the Common Ratio (r).
Example: 3, 6, 12, 24...
The ratio (r) is 2 (because 3 * 2 = 6).
The General Formula:
4. What is a Series?
While a sequence is a list, a Series is the sum of that list. We use the symbol Sn to represent the sum of the first 'n' terms.
- Sequence: 2, 4, 6
- Series: 2 + 4 + 6 = 12
Sigma Notation (Summation Notation)
Mathematicians use the Greek letter Sigma ($\Sigma$) to write long sums compactly.
If you see $\Sigma$ with $n=1$ on the bottom and $5$ on top, it means "Plug in 1, then 2, then 3... up to 5, and add them all up."
5. Summing Arithmetic Series
There is a famous story about the mathematician Gauss. As a child, his teacher asked the class to add numbers 1 to 100. Gauss answered instantly: 5050. How?
He realized that the first and last numbers pair up to the same value (1+100=101, 2+99=101). He just needed to multiply that pair sum by half the number of terms.
6. Summing Geometric Series
Summing geometric series (like 1 + 2 + 4 + 8...) requires a different formula because the numbers grow exponentially.
7. Infinite Geometric Series
This is where math gets weird. Can you add an infinite list of numbers and get a real answer?
Yes! But only if the numbers are getting smaller (specifically, if -1 < r < 1).
Consider: 1/2 + 1/4 + 1/8 + 1/16...
You are always adding half of what remains. You will get closer and closer to 1, but never pass it. We say this series converges to 1.
The Infinite Sum Formula:
This formula is crucial for engineering and fractals.
8. Conclusion
Sequences and Series allow us to predict the future behavior of a pattern and calculate totals without doing endless addition. Whether you are dealing with linear progression (Arithmetic) or exponential growth (Geometric), these formulas are the tools that let you handle lists of data efficiently.