Exercise 1: Limit of a Sequence and the Golden Ratio (Minus 1)

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2025-04-14 04:29

In this video I show that the limit of a sequence is the same if the sequence was shifted by 1 term. I use this fact to solve for the limit of a recursive sequence by plugging in the limit into the recursion relation itself. Using the definition of a limit, shifting n to n + 1 doesn't fundamentally change the limit, thus the limits are the same. The recursive sequence I look at starts at 1 and has a recursion relation 1 / (1 + previous term). Solving this limit yields a quadratic equation, and solving this for the positive limit term yields THE golden ratio but subtracted by one: 0.6180339... Amazing stuff!

#math #sequences #goldenratio #education #calculus

Timestamps:

Exercise 1: 0:00
Solution to Part (a): 0:30
- Replacing n with n + 1 in the definition of a limit: 2:02
Solution to Part (b): Recursive sequence: 4:30
- Plugging in the limit into the recursion relation: 6:22
- Recall the Quadratic equation: 7:14
- Solving the limit using Quadratic formula: 7:55
- Limit is positive so take the positive solution: 9:35
- Solution is the golden ratio phi minus one: https://en.wikipedia.org/wiki/Golden_ratio 9:45