Fly Fishing For Dummies

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what is the long term number of fish?

a0=300 fish. decreases by 15%, add 50.

a1=95
a3=2514 fish

how do i find the long term value (the limit)?

make it for dummies, math flies right over my head...

I assume that "decreases by 15%, add 50" is the rule to find the number a(n+1) of fish in the next generation from the number of fish a(n) in the generation before. A decrease by 15% means that 85% or a fraction of 85/100 remains, so that:

a(n+1) = (85/100) a(n) + 50.

However, then I cannot make sense of a1=95, a3=2514 fish. I get:
a0 = 300
a1 = 305
a2 = 309.25
a3 = 312.8625

To find the limit (if it exists), let's give it a name: L. If a(n) gets very close to L, then so does a(n+1), so let us replace both by L in the equation:

L = (85/100) L + 50.

Solving for L gives L = 1000/3 = 333.333...

Wrightsville Beach Fishing & Fly Tying, The Spoon Fly


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