Question

You buy a bag of skittles that contains 500 pieces of candy in it. Each day you eathalf of the amount that is in the bag.1.Write an explicit and recursive formula for this sequence2.How many days will it take to where there is less than one skittle in the bag?

Answer 1

1. the sequence is:

500, 250, 125, ...

the common ratio (r) is:

250/500 = 1/2 = 0.5

125/250 = 1/2 = 0.5

So, the sequence is a geometric sequence.

Explicit formula:

$\begin{gathered} a_n=a_1\cdot r^{n\text{ - 1}} \\ a_n=500\cdot0.5^{n\text{ - 1}} \end{gathered}$

Recursive formula:

$\left\{ \begin{aligned}a_1=500 \\ a_n=r\cdot a_{n\text{ - 1}}\end{aligned}\right.$

2. In the formulas, n represents days. Using the explicit formula with an = 1, we get:

$\begin{gathered} 1\text{ = 500}\cdot0.5^{n\text{ - 1}} \\ \frac{1}{500}=0.5^{n\text{ - 1}} \\ ln(\frac{1}{500})\text{ = (n - 1)}\cdot\ln (0.5) \\ \frac{ln(\frac{1}{500})}{ln(0.5)}+1\text{ = n} \\ 9.965=\text{ n} \end{gathered}$

then, after 10 days there will be less than 1 skittle in the bag