Since each day, the amount of candy that Augustus will give away is 60% of the actual amount of candy, the amount of candy that has on day n depends on the amount of candy remaining from the day n-1. Let C_n be the amount of candy remaining on day n.
Since 60% of the candy from the day n-1 will be given away, then only 40% from the candy of the day n-1 will remain on day n. Then:
[tex]\begin{gathered} C_n=\frac{40}{100}\times C_{n-1} \\ =0.4C_{n-1} \end{gathered}[/tex]
Since the amount of candy on day 1 is 100,000, then the recursive formula is:
[tex]\begin{gathered} C_n=0.4C_{n-1} \\ C_1=100,000 \end{gathered}[/tex]
After n-1 days, the initial amount of candies gets multiplied by a factor of 0.4 n-1 times. Then, the explicit formula for the amount of candies that remain on day n (after n-1 days) is:
[tex]\begin{gathered} C_n=(0.4)^{n-1}\times C_1 \\ =0.4^{n-1}\times100,000 \end{gathered}[/tex]
Therefore, the answers are:
Explicit formula:
[tex]C_n=0.4^{n-1}\times100,000[/tex]
Recursive formula:
[tex]\begin{gathered} C_n=0.4\times C^{}_{n-1} \\ C_1=100,000 \end{gathered}[/tex]
