Write an equation to define sequence k recursivley.

Question

A piece of paper has an area of 81 cm2. A strip is cut off that is \small \frac{1}{3} the original area. From that strip, another strip is cut off that is \small \frac{1}{3} the area of the first, and so on.

Here is a graph and table representing sequence k, where k(n) is the area in square centimeters of the strip of paper after n cuts.

Answer 1

It is a geometric progression with the common ratio of 1/3 (or 3^-1).

k_n = k₁ • (1/3)^n-1 ← formula for the n-th term of the progression

k_n - the n-th term of the progression

k₁ - the 1-st term of the progression ← k₁ = 81

Note that 81 = 3⁴

Therefore, the progression is

3⁴, 3³, 3², 3¹, 3⁰, 3^-1, 3^-2, 3^-3, etc.