First, let's observe the given sequence:
[tex]\[ \frac{1}{2}, 1, \frac{9}{8}, 1, \frac{25}{32} \][/tex]
To find the next terms in the sequence, we'll identify any patterns.
### Step 1: Identify the Pattern
1. The sequence alternates between the value [tex]\(1\)[/tex] and fractions that follow some discernible pattern.
2. For the fractions, observe how their numerators and denominators change:
- [tex]\(\frac{1}{2}\)[/tex] has numerator [tex]\(1 = 1^2\)[/tex] and denominator [tex]\(2 = 2^1\)[/tex]
- [tex]\(\frac{9}{8}\)[/tex] has numerator [tex]\(9 = 3^2\)[/tex] and denominator [tex]\(8 = 2^3\)[/tex]
- [tex]\(\frac{25}{32}\)[/tex] has numerator [tex]\(25 = 5^2\)[/tex] and denominator [tex]\(32 = 2^5\)[/tex]
The numerators are squares of odd numbers [tex]\(1, 3, 5,\)[/tex] and so on.
The denominators are powers of 2, corresponding to the exponent of the odd squares. For example, [tex]\(9 = 3^2 \Rightarrow 2^3\)[/tex] and [tex]\(25 = 5^2 \Rightarrow 2^5\)[/tex].
### Step 2: Calculate the Next Terms
The Next Term in the Alternation
Since the sequence alternates and the last term observed is a fraction [tex]\(\frac{25}{32}\)[/tex], the next term in the sequence must be [tex]\(1\)[/tex].
### Step 3: Calculate the Subsequent (Fraction) Term
1. Following the pattern, the next odd number after [tex]\(5\)[/tex] is [tex]\(7\)[/tex].
2. Square this number: [tex]\(7^2 = 49\)[/tex].
3. The exponent of [tex]\(2\)[/tex] should match the odd number sequence in the pattern, so the denominator will be [tex]\(2^7\)[/tex]:
- [tex]\(2^7 = 128\)[/tex].
Combine these:
[tex]\[ \text{Next fraction} = \frac{7^2}{2^7} = \frac{49}{128} \][/tex]
### Step 4: Write the Solution
Finally, summarizing the pattern and terms added to the sequence:
The sequence is:
[tex]\[ \frac{1}{2}, 1, \frac{9}{8}, 1, \frac{25}{32}, 1, \frac{49}{128} \][/tex]
So, the next two terms are:
[tex]\[ 1 \quad \text{and} \quad \frac{49}{128} \][/tex]
Thus the sequence extended becomes:
[tex]\[ \frac{1}{2}, 1, \frac{9}{8}, 1, \frac{25}{32}, 1, \frac{49}{128} \][/tex]
