To find the missing rational number in the series [tex]\(\frac{4}{9}, \frac{9}{20}, \ldots, \frac{45}{100}\)[/tex], we need to observe patterns in both the numerators and denominators.
### Step 1: Observing the numerators
The given numerators are:
1. 4 in the fraction [tex]\(\frac{4}{9}\)[/tex]
2. 9 in the fraction [tex]\(\frac{9}{20}\)[/tex]
3. ? (missing numerator)
4. 45 in the fraction [tex]\(\frac{45}{100}\)[/tex]
Let's find the missing numerator by observing a possible pattern in the given numerators.
The numerators (4, 9, ?, 45) suggest an increasing sequence. One obvious guess is to see if they fit a polynomial or a simple number sequence pattern.
Notice the pattern:
- [tex]\(4 = 2^2\)[/tex]
- [tex]\(9 = 3^2\)[/tex]
- [tex]\(45 = 45\)[/tex] (actually [tex]\(45 = 5 \times 9\)[/tex])
Upon closer inspection, it seems they do not follow a simple square pattern, but they could follow a sequence.
### Step 2: Observing the denominators
The given denominators are:
1. 9 in the fraction [tex]\(\frac{4}{9}\)[/tex]
2. 20 in the fraction [tex]\(\frac{9}{20}\)[/tex]
3. ? (missing denominator)
4. 100 in the fraction [tex]\(\frac{45}{100}\)[/tex]
The denominators (9, 20, ?, 100) also need to be examined for a pattern.
### Step 3: Infer the Patterns
To better infer the intermediate term, we can consider both numerators and denominators together:
- Look for patterns like arithmetic sequence, geometric sequence or fractional relationship changes.
One approach could be to assume linear relationships.
#### Numerator Pattern:
- The difference between the numerators are:
[tex]\(9 - 4 = 5\)[/tex],
[tex]\(45 - 9 = 36\)[/tex]
- We might interpolate the mid-point numerical value between an increasing gap.
#### Denominator Pattern:
- The difference between denominators are:
[tex]\(20 - 9 = 11\)[/tex],
[tex]\(100 - 20 = 80\)[/tex]
Another approach is the fractional mean:
- For numerators: [tex]\((4 + 45)/2 = 24.5\)[/tex]
- For denominators: [tex]\((9 + 100)/2 = 54.5\)[/tex].
However, since fractions only work in rational terms, and the series seems arithmetic-like yet large differences, other plausible interpolations are:
Let’s test with simpler middle terms. Notice potential:
- Both satisfy approximate equal splits in terms of scaling similarity.
We should refine:
- Numerator precise intermediate: [tex]\(24 \implies 17.5\)[/tex]; simplified for cleaner pairs closer to natural doubling would produce: [tex]\(((9)+ (20))/2) approx \(65\)[/tex] divisible simpler scaled thus correspondence fraction:
Intermediate suggests:
[tex]\[\boxed{\frac{21}{60}}\][/tex]
Thus: Exact rational missing term predicted is [tex]\(\frac{21}{58}?\)[/tex]
