Exploring the logic of sequences, from the growth of triangular numbers to the infinite self-similarity of fractals.
A sequence is an ordered list of numbers where each number is called a term. Patterns help us predict future terms and solve real-life problems.
1, 3, 6, 10, 15...
1, 4, 9, 16, 25...
Sum of first n odd numbers.
We use \( t_n \) to represent the term in the \( n^{th} \) position. Subscripts match the term numbers!
Uses the position \( n \) directly to find the value.
\( t_n = 2n - 1 \) (Odds)
Relates a term to previous terms.
\( t_n = t_{n-1} + 3 \)
Studied by Virahānka (7th Century CE) in the context of Prakrit poetry meter! Each term is the sum of the previous two.
1, 2, 3, 5, 8, 13...
An AP is a sequence where the difference between consecutive terms is constant (\( d \)).
When plotted on a graph, the points of an AP lie exactly on a straight line.
Discovered by Āryabhaṭa in the 5th Century, this formula allows us to sum hundreds of numbers instantly.
Verse 19, Chapter 2: The sum is the average of the first and last terms multiplied by the number of terms.
In a GP, each term is obtained by multiplying the previous term by a fixed ratio (\( r \)).
The Sierpiński Triangle shows GP in action: Number of triangles increases by factor 3, while Area decreases by factor 3/4.
A bouncing ball reaching 75% height each time forms a decreasing GP.