To determine the next three numbers in the given pattern, observe the operations involved in transitioning from one number to the next within the sequence. Here is the given sequence:
[tex]\[ 25, 5, 1, \frac{1}{5}, \frac{1}{25}, \frac{1}{50}, \frac{1}{100} \][/tex]
Let's analyze the pattern in detail:
1. From 25 to 5:
[tex]\[ 25 \div 5 = 5 \][/tex]
2. From 5 to 1:
[tex]\[ 5 \div 5 = 1 \][/tex]
3. From 1 to [tex]\(\frac{1}{5}\)[/tex]:
[tex]\[ 1 \div 5 = \frac{1}{5} \][/tex]
4. From [tex]\(\frac{1}{5}\)[/tex] to [tex]\(\frac{1}{25}\)[/tex]:
[tex]\[ \frac{1}{5} \div 5 = \frac{1}{25} \][/tex]
Notice that so far, each term has been divided by 5. The pattern shifts after this:
5. From [tex]\(\frac{1}{25}\)[/tex] to [tex]\(\frac{1}{50}\)[/tex]:
[tex]\[ \frac{1}{25} \div 2 = \frac{1}{50} \][/tex]
6. From [tex]\(\frac{1}{50}\)[/tex] to [tex]\(\frac{1}{100}\)[/tex]:
[tex]\[ \frac{1}{50} \div 2 = \frac{1}{100} \][/tex]
After the first four divisions by 5, subsequent terms are being divided by 2. Following this pattern, we now need to continue dividing by 2 to find the next three terms.
Starting with the last number in the sequence:
[tex]\[ \frac{1}{100} \][/tex]
The next number in the pattern will be:
1. Dividing [tex]\(\frac{1}{100}\)[/tex] by 2:
[tex]\[ \frac{1}{100} \div 2 = \frac{1}{200} \][/tex]
2. Next, divide [tex]\(\frac{1}{200}\)[/tex] by 2:
[tex]\[ \frac{1}{200} \div 2 = \frac{1}{400} \][/tex]
3. Next, divide [tex]\(\frac{1}{400}\)[/tex] by 2:
[tex]\[ \frac{1}{400} \div 2 = \frac{1}{800} \][/tex]
To write these as decimal values for clarity:
- [tex]\(\frac{1}{200} = 0.005\)[/tex]
- [tex]\(\frac{1}{400} = 0.0025\)[/tex]
- [tex]\(\frac{1}{800} = 0.00125\)[/tex]
Hence, the next three numbers in the pattern are:
[tex]\[ 0.005, 0.0025, 0.00125 \][/tex]
