Consider the decimal 0.27227222722227... and answer the following questions.

Question

1. Is the decimal number rational or​ irrational? Explain.
2. Assuming the pattern​ continues, what are the next 6 digits in the decimal​ number?
3. How many times will the digit 2 appear in total before the sixth 7 in the decimal​ number?
4. The digit 7 appears in the second decimal​ place, the fifth decimal​ place, and the ninth decimal place. What are the next three places in which the digit 7​ appears? Describe this pattern.

Answer 1

Let's analyze the decimal number 0.27227222722227...

1. Is the decimal number rational or​ irrational? Explain.

The decimal number 0.27227222722227... is rational because it is a repeating decimal. The sequence "272" repeats, with increasingly more "2"s appearing before each "7". Since it can be expressed as a fraction (even though it's a bit complex), it is classified as a rational number.

2. Assuming the pattern continues, what are the next 6 digits in the decimal​ number?

Looking at the pattern, we observe:

• The first segment is "272"
• The second segment is "2722"
• The third segment is "27222"
• The fourth segment is "272222"...

Following this pattern, the next segment should be "2722222".

Thus, the next 6 digits after "27227222722227..." are: 272222

3. How many times will the digit 2 appear in total before the sixth 7 in the decimal​ number?

We can analyze the number of "2"s that appear up to the sixth occurrence of "7".

Counting the "7"s:

1. 2nd decimal place - 7
2. 5th decimal place - 7
3. 9th decimal place - 7
4. 12th decimal place - 7
5. 15th decimal place - 7
6. 18th decimal place - 7

We have reached the sixth "7" in the 18th decimal place.

Now let’s count the "2"s up to the 18th decimal place: Following the provided pattern:

• 1st occurrence: 2 (total: 1)
• 2nd occurrence: 7 followed by one "2" (total: 2)
• 3rd occurrence: 2 followed by "2722" (total: 4)
• 4th occurrence: 2 followed by "27222" (total: 6)
• 5th occurrence: 2 followed by "272222" (total: 8)

Counting the occurrences of "2" explicitly:

• 1st "272": 2
• 2nd "2722": 3
• 3rd "27222": 4
• 4th "272222": 5

From these segments, in total up to the sixth "7":

• 1st: 2
• 2nd: 3
• 3rd: 4
• 4th: 5

Total = 2 + 3 + 4 + 5 = 14.

So, the number of times the digit 2 will appear before the sixth 7 is 14 times.

4. The digit 7 appears in the second decimal place, the fifth decimal place, and the ninth decimal place. What are the next three places in which the digit 7​ appears? Describe this pattern.

As analyzed:

1. The first "7" appears at decimal place 2.
2. The second "7" appears at decimal place 5.
3. The third "7" appears at decimal place 9.

Notice that the positions of the "7"s are increasing by an additional value each time:

• From 2 to 5 (increased by 3)
• From 5 to 9 (increased by 4)

The pattern seems to suggest that the next increase would be by 5:

• From 9, increasing by 5 gives: 9 + 5 = 14 (4th "7")
• Next increase would be by 6: 14 + 6 = 20 (5th "7")
• Next increase would be by 7: 20 + 7 = 27 (6th "7")

Thus, the next three places in which the digit 7 appears are:

• 14, 20, and 27.

Summary:

1. The decimal is rational.
2. The next 6 digits are 272222.
3. The digit "2" appears 14 times total before the sixth "7".
4. The next places for "7" are 14, 20, and 27.