Answer :
Certainly! Let’s carefully analyze and solve this problem step by step:
### Part (a): Describe the Pattern and Write the Next Three Numbers
Pattern Analysis:
Given numbers: [tex]\(1, 1 \frac{1}{2}, 1 \frac{3}{4}, 1 \frac{7}{8}, 1 \frac{15}{16}\)[/tex]
1. Convert each mixed number to an improper fraction:
[tex]\[ 1 = 1,\quad 1\frac{1}{2} = 1 + \frac{1}{2} = 1.5, \quad 1\frac{3}{4} = 1 + \frac{3}{4} = 1.75, \quad 1\frac{7}{8} = 1 + \frac{7}{8} = 1.875, \quad 1\frac{15}{16} = 1 + \frac{15}{16} = 1.9375 \][/tex]
2. Observe the fractional parts:
[tex]\(\frac{1}{2}, \frac{3}{4}, \frac{7}{8}, \frac{15}{16}\)[/tex]
3. Identify the pattern:
- The denominators are powers of 2: [tex]\(2, 4, 8, 16\)[/tex].
- The numerators follow the pattern of being one less than a power of 2: [tex]\(1 = 2^1-1, 3 = 2^2-1, 7 = 2^3-1, 15 = 2^4-1\)[/tex].
Next Three Numbers:
Following the identified pattern:
- The next fraction's numerator should be [tex]\(2^5 - 1 = 31\)[/tex] and the denominator should be [tex]\(2^5 = 32\)[/tex], giving the fraction [tex]\(\frac{31}{32}\)[/tex]. Therefore, the next number is [tex]\(1 \frac{31}{32}\)[/tex].
- The following number will have the numerator [tex]\(2^6 - 1 = 63\)[/tex] and the denominator [tex]\(2^6 = 64\)[/tex], giving the fraction [tex]\(\frac{63}{64}\)[/tex]. Thus, the next number is [tex]\(1 \frac{63}{64}\)[/tex].
- The subsequent number will have the numerator [tex]\(2^7 - 1 = 127\)[/tex] and the denominator [tex]\(2^7 = 128\)[/tex], giving the fraction [tex]\(\frac{127}{128}\)[/tex]. Consequently, the next number is [tex]\(1 \frac{127}{128}\)[/tex].
### Next three numbers: [tex]\(1 \frac{31}{32}, 1 \frac{63}{64}, 1 \frac{127}{128}\)[/tex]
Converting these to decimals:
[tex]\[ 1 \frac{31}{32} = 1 + \frac{31}{32} = 1.96875, \quad 1 \frac{63}{64} = 1 + \frac{63}{64} = 1.984375, \quad 1 \frac{127}{128} = 1 + \frac{127}{128} = 1.9921875 \][/tex]
### Part (b): What is Happening to the Values of the Numbers?
As observed, the values of the numbers are getting closer and closer to 2. The series of numbers converges to 2 because the fractional part is increasing but always remains less than 1, thereby making the entire number approach 2 from below.
### Part (c): Conjecture About Later Numbers
Based on the observed pattern, as the sequence progresses, the fractional part will continue to get closer to 1, since the numerator will be just one less than the denominator, which is an increasing power of 2.
Conjecture:
As [tex]\( n \)[/tex] approaches infinity, the values of the numbers in this sequence will get closer and closer to 2, but will never actually reach 2. This happens because each fraction added to 1 is of the form [tex]\(\frac{2^n - 1}{2^n}\)[/tex], which approaches 1 as [tex]\( n \to \infty \)[/tex].
### Conclusion:
The sequence [tex]\(1, 1 \frac{1}{2}, 1 \frac{3}{4}, 1 \frac{7}{8}, 1 \frac{15}{16}, 1 \frac{31}{32}, 1 \frac{63}{64}, \ldots\)[/tex] exhibits a clear pattern where each term gets progressively closer to 2, based on the property of fractional parts constructed from increasing powers of 2.
### Part (a): Describe the Pattern and Write the Next Three Numbers
Pattern Analysis:
Given numbers: [tex]\(1, 1 \frac{1}{2}, 1 \frac{3}{4}, 1 \frac{7}{8}, 1 \frac{15}{16}\)[/tex]
1. Convert each mixed number to an improper fraction:
[tex]\[ 1 = 1,\quad 1\frac{1}{2} = 1 + \frac{1}{2} = 1.5, \quad 1\frac{3}{4} = 1 + \frac{3}{4} = 1.75, \quad 1\frac{7}{8} = 1 + \frac{7}{8} = 1.875, \quad 1\frac{15}{16} = 1 + \frac{15}{16} = 1.9375 \][/tex]
2. Observe the fractional parts:
[tex]\(\frac{1}{2}, \frac{3}{4}, \frac{7}{8}, \frac{15}{16}\)[/tex]
3. Identify the pattern:
- The denominators are powers of 2: [tex]\(2, 4, 8, 16\)[/tex].
- The numerators follow the pattern of being one less than a power of 2: [tex]\(1 = 2^1-1, 3 = 2^2-1, 7 = 2^3-1, 15 = 2^4-1\)[/tex].
Next Three Numbers:
Following the identified pattern:
- The next fraction's numerator should be [tex]\(2^5 - 1 = 31\)[/tex] and the denominator should be [tex]\(2^5 = 32\)[/tex], giving the fraction [tex]\(\frac{31}{32}\)[/tex]. Therefore, the next number is [tex]\(1 \frac{31}{32}\)[/tex].
- The following number will have the numerator [tex]\(2^6 - 1 = 63\)[/tex] and the denominator [tex]\(2^6 = 64\)[/tex], giving the fraction [tex]\(\frac{63}{64}\)[/tex]. Thus, the next number is [tex]\(1 \frac{63}{64}\)[/tex].
- The subsequent number will have the numerator [tex]\(2^7 - 1 = 127\)[/tex] and the denominator [tex]\(2^7 = 128\)[/tex], giving the fraction [tex]\(\frac{127}{128}\)[/tex]. Consequently, the next number is [tex]\(1 \frac{127}{128}\)[/tex].
### Next three numbers: [tex]\(1 \frac{31}{32}, 1 \frac{63}{64}, 1 \frac{127}{128}\)[/tex]
Converting these to decimals:
[tex]\[ 1 \frac{31}{32} = 1 + \frac{31}{32} = 1.96875, \quad 1 \frac{63}{64} = 1 + \frac{63}{64} = 1.984375, \quad 1 \frac{127}{128} = 1 + \frac{127}{128} = 1.9921875 \][/tex]
### Part (b): What is Happening to the Values of the Numbers?
As observed, the values of the numbers are getting closer and closer to 2. The series of numbers converges to 2 because the fractional part is increasing but always remains less than 1, thereby making the entire number approach 2 from below.
### Part (c): Conjecture About Later Numbers
Based on the observed pattern, as the sequence progresses, the fractional part will continue to get closer to 1, since the numerator will be just one less than the denominator, which is an increasing power of 2.
Conjecture:
As [tex]\( n \)[/tex] approaches infinity, the values of the numbers in this sequence will get closer and closer to 2, but will never actually reach 2. This happens because each fraction added to 1 is of the form [tex]\(\frac{2^n - 1}{2^n}\)[/tex], which approaches 1 as [tex]\( n \to \infty \)[/tex].
### Conclusion:
The sequence [tex]\(1, 1 \frac{1}{2}, 1 \frac{3}{4}, 1 \frac{7}{8}, 1 \frac{15}{16}, 1 \frac{31}{32}, 1 \frac{63}{64}, \ldots\)[/tex] exhibits a clear pattern where each term gets progressively closer to 2, based on the property of fractional parts constructed from increasing powers of 2.
