Question

A piece of paper has an area of 81 cm².A strip is cut off that is 1/3 the original area. From that strip, another strip is cut off that is 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.Write a recursive definition for this table

Answer 1

ANSWER

$a_n=a_{n-1}\cdot\text{ }\frac{1}{3}$

EXPLANATION

We want to write the recursive definition of the situation represented in the table.

The original area of the paper is 81 cm² and with each strip that is cut off, we are left with 1/3 of the area of the former paper.

We can therefore, say that this situation represents a geometric progression, where the value of the next term can be gotten by multiplying a constant factor to the former term.

The general recursive definition of a geometric progression is given as:

$\begin{gathered} a_n=a_{n\text{ - 1}}\cdot\text{ r} \\ _{}where\text{ an is the nth term} \\ a(n\text{ - 1) is the (n - 1)th term or the term before the nth term} \\ r\text{ = common ratio} \end{gathered}$

The common ratio is the factor that multiplies each term, as described earlier.

From the question, the common ratio is 1/3, since each new strip is 1/3 the area of the former strip.

Therefore, the recursive definition of the data in the table is:

$a_n=a_{n-1}\cdot\text{ }\frac{1}{3}$