Answer:
On day 9, there will be only one apple on the tree.
Step-by-step explanation:
Geometric Sequence
In the geometric sequences, each term is found by multiplying (or dividing) the previous term by a fixed number, called the common ratio.
On day 1, there are 256 apples on the tree.
On the next day, there are half of the apples: 256/2=128 apples
On the next day, there are half of the apples: 128/2=64 apples
This is a geometric sequence with a common ratio of 1/2.
The general formula for the nth term of a geometric sequence is:
[tex]a_n=a_1*r^{n-1}[/tex]
Where a1 is the first term and r is the common ratio. We need to find the value of n that results in only one apple in the tree:
[tex]\displaystyle 1=256*\left(\frac{1}{2}\right)^{n-1}[/tex]
Dividing by 256:
[tex]\displaystyle \frac{1}{256}=\left(\frac{1}{2}\right)^{n-1}[/tex]
Since [tex]256=2^8[/tex]:
[tex]\displaystyle \frac{1}{2^8}=\left(\frac{1}{2}\right)^{n-1}[/tex]
Applying exponents property:
[tex]\frac{1}{2^8}=\left(\frac{1}{2}\right)^{8}[/tex]
[tex]\displaystyle \left(\frac{1}{2}\right)^{8}=\left(\frac{1}{2}\right)^{n-1}[/tex]
Equating the exponents:
n - 1 = 8
n = 9
On day 9, there will be only one apple on the tree.
