Question

f(x) = 18*(2/3)^(x-1), for x = 1, 2, 3, ...

(Each step multiplies by 2/3, and at x=1 the value is 18.)

Answer 1

It’s a geometric sequence with first term a1 = 18 and common ratio r = 2/3 (domain x = 1,2,3,...).

Useful facts:
- General term: f(x) = 18*(2/3)^(x-1).
- Recursive relation: f(x+1) = (2/3) f(x).
- First few values:
- f(1) = 18
- f(2) = 12
- f(3) = 8
- f(4) = 16/3 ≈ 5.333
- f(5) = 32/9 ≈ 3.556
- f(6) = 64/27 ≈ 2.370
- Sum of first n terms: S_n = 18 * (1 - (2/3)^n) / (1 - 2/3) = 54 * (1 - (2/3)^n).
- Infinite sum (as n → ∞): S∞ = 18 / (1 - 2/3) = 54.
- Limit as x → ∞: f(x) → 0.

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