Geometric Series & Special Sequences
Geometric series follow the same idea as arithmetic series, but for geometric sequences. Use Sₙ = a₁1−rn/1−r to find the sum of the first n terms.
Here's something mind-blowing: if your common ratio r is between -1 and 1 like1/2or1/3, you can find the sum of an infinite geometric series using S∞ = a₁/1−r. This means some infinite sequences actually add up to a finite number!
Harmonic sequences are created by taking reciprocals of arithmetic sequences. So if you have 1, 3, 5, 7..., the harmonic sequence would be 1/1, 1/3, 1/5, 1/7...
The Fibonacci sequence is probably the most famous sequence in math: 1, 1, 2, 3, 5, 8, 13... Each term equals the sum of the two terms before it. You'll find Fibonacci numbers everywhere in nature!
Fun Fact: The Fibonacci sequence appears in flower petals, shell spirals, and even your family tree!