There are exactly 50 odd numbers that lie between 100 and 200.
This specific count is derived by identifying the range of odd numbers and applying principles of arithmetic progression, a method also supported by expert explanations.
When asked for numbers "between 100 and 200," it implies that we exclude 100 and 200 themselves. Therefore, we are looking for odd numbers starting from the first one greater than 100 and ending with the last one less than 200.
Identifying the First and Last Odd Numbers
- The first odd number immediately following 100 is 101.
- The last odd number immediately preceding 200 is 199.
So, the sequence of odd numbers we are interested in is 101, 103, 105, ..., 199.
Calculating the Count of Odd Numbers
Using the Arithmetic Progression Formula
Odd numbers form an arithmetic progression (AP) where each subsequent term increases by a fixed value. In this sequence (101, 103, 105, ...), the common difference (d) between consecutive terms is 2 (e.g., 103 - 101 = 2).
We can use the formula for the nth term of an AP:
$a_n = a_1 + (n - 1)d$
Where:
- $a_n$: the last term in the sequence (199)
- $a_1$: the first term in the sequence (101)
- $d$: the common difference (2)
- $n$: the number of terms (what we want to find)
Let's plug in the values and solve for n:
| Variable | Value | Description |
|---|---|---|
| $a_n$ | 199 | Last odd number in the range |
| $a_1$ | 101 | First odd number in the range |
| $d$ | 2 | Common difference between odd numbers |
| $n$ | ? | Number of odd numbers (to be found) |
$199 = 101 + (n - 1)2$
$199 - 101 = (n - 1)2$
$98 = (n - 1)2$
$98 / 2 = n - 1$
$49 = n - 1$
$n = 49 + 1$
$n = 50$
Therefore, there are 50 odd numbers between 100 and 200.
Quick Calculation Method
A simpler and often quicker way to determine the count is by understanding the distribution of odd and even numbers within a given range:
- Total odd numbers up to 200: In any sequence of consecutive integers starting from 1, half of the numbers are odd and half are even. So, from 1 to 200, there are $200 / 2 = 100$ odd numbers (1, 3, ..., 199).
- Odd numbers up to 100: Similarly, from 1 to 100, there are $100 / 2 = 50$ odd numbers (1, 3, ..., 99).
- Odd numbers between 100 and 200: These are the odd numbers from 101 to 199. To find their count, we subtract the odd numbers up to 100 from the total odd numbers up to 200:
$100 (\text{odd numbers up to 200}) - 50 (\text{odd numbers up to 100}) = 50$.
Expert Confirmation and Summary
As highlighted by an Expert-Verified Answer from Brainly.in regarding this specific question:
"Common difference d = second term - first term" = 103 - 101 = 2. n = 50, The number of odd numbers between 100 and 200 are 50."
This confirms our calculation and understanding.
In summary:
- First odd number in range: 101
- Last odd number in range: 199
- Total count: 50
Whether calculated using the arithmetic progression formula or by simple subtraction of counts within ranges, the result consistently shows that there are precisely 50 odd numbers between 100 and 200.
