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.
Question content area bottom
Part 1
1. Choose the correct answer below.
A.
The decimal number is irrational because there is no pattern in the digits of the decimal.
B.
The decimal number is rational because it is composed of only 2s and 7s.
C.
The decimal number is irrational because even though the decimal digits follow a​ pattern, the decimal is not a repeating decimal.
This is the correct answer.D.
The decimal number is rational because there is a clear pattern in the decimal digits with the number of 2s before each 7.
Your answer is not correct.
Part 2
2. The next 6 digits in the decimal number are
  
222227.
​(Type a whole​ number.)
Part 3
3. The digit 2 will appear
  
21 times in total before the sixth 7 in the decimal number.
​(Type a whole​ number.)
Part 4
4. What are the next three places in which the digit 7​ appears?
A.
​21st, 29th, 37th
B.
​15th, 21st, 22nd
C.
​19th, 26th, 34th
Your answer is not correct.D.
​20th, 27th, 35th
This is the correct answer.
Part 5
Describe the pattern.
From the first appearance of 7 in the
▼
​position, the next 7 is
  
enter your response here places to the right. The next 7 is
  
enter your response here places to the​ right, then
  
enter your response here ​places, then
  
enter your response here ​places, and so on.
​(Type whole​ numbers.)

Answer 1

Let's break down each part of the question step by step:

Part 1: Rational or Irrational?

1. Is the decimal number rational or​ irrational?
   • The correct answer is C: The decimal number is irrational because even though the decimal digits follow a​ pattern, the decimal is not a repeating decimal.

Part 2: Next 6 Digits

2. Assuming the pattern​ continues, what are the next 6 digits in the decimal​ number?
   • The given pattern is that each 7 is followed by an increasing number of 2s: 1 two, then 2 twos, then 3 twos, and so forth.
   • After the last "22227", the next six digits would be 222227.

Part 3: Total Count of 2s Before the Sixth 7

3. How many times will the digit 2 appear in total before the sixth 7 in the decimal number?
   • The occurrences of the digit "2" before the sixth "7":
     • 1st 7: 1 two
     • 2nd 7: 2 twos
     • 3rd 7: 3 twos
     • 4th 7: 4 twos
     • 5th 7: 5 twos
     • 6th 7: 6 twos
   • The total 2s before the 6th 7 can be summed up:
     • \( 1 + 2 + 3 + 4 + 5 = 15 \)
   • Thus, before the sixth 7, the digit "2" will appear 15 times.

Part 4: Next Occurrences of Digit 7

4. What are the next three places in which the digit 7​ appears?
   • The digit 7 appears in a recurring pattern where:
     • The first occurrence was at the 2nd position.
     • The second occurrence was at the 5th position.
     • The third at the 9th position.
     • Following this logic, we find the next instances likely at positions:
       • After 1st (2), there are 1 two
       • After 2nd (7), there are 2 twos therefore the next 7 would be at position 5 (2 + 3)
       • After 3rd (7) there are 3 twos, leading to position 9 (5 + 4)
       • Continuing this pattern would suggest the next positions are 20th, 27th, 35th.
   • Hence, the correct choice is D: 20th, 27th, 35th.

Part 5: Describe the Pattern

5. Describe the pattern for 7s appearing.
   • From the first appearance of 7 in the 2nd position, the next 7 is 3 places to the right. The next 7 is 4 places to the right, then 5 places, and so on. Thus:
     • The first increment is 3, followed by 4, then 5, and so forth.

To summarize, the following answers can be consolidated:

1. C: The decimal is irrational.
2. 222227: Next 6 digits.
3. 15 times: Digit 2 appears.
4. 20th, 27th, 35th: Next occurrences for digit 7.
5. Describing the increments: "From the first appearance of 7 in the 2nd position, the next 7 is 3 places to the right. The next 7 is 4 places, then 5 places, and so on."