If {x} is the least integer greater than or equal to x, then find the value of the following series:
`{sqrt1}+{sqrt2}+{sqrt3}+{sqrt4}+...+{sqrt(99)}+{sqrt(100)}`

Options

A. 715
B. 55
C. 157
D. 835

Answer

To solve the problem, we need to evaluate the series:

\[ \{ \sqrt{1} \} + \{ \sqrt{2} \} + \{ \sqrt{3} \} + \{ \sqrt{4} \} + \ldots + \{ \sqrt{100} \} \]

where \(\{x\}\) denotes the least integer greater than or equal to \(x\).

### Step-by-Step Solution:

1. **Identify the values of \(\{ \sqrt{n} \}\)**:
- For \(n = 1\): \(\{ \sqrt{1} \} = \{ 1 \} = 1\)
- For \(n = 2\): \(\{ \sqrt{2} \} \approx 1.414 \Rightarrow \{ \sqrt{2} \} = 2\)
- For \(n = 3\): \(\{ \sqrt{3} \} \approx 1.732 \Rightarrow \{ \sqrt{3} \} = 2\)
- For \(n = 4\): \(\{ \sqrt{4} \} = \{ 2 \} = 2\)
- For \(n = 5\): \(\{ \sqrt{5} \} \approx 2.236 \Rightarrow \{ \sqrt{5} \} = 3\)
- For \(n = 6\): \(\{ \sqrt{6} \} \approx 2.449 \Rightarrow \{ \sqrt{6} \} = 3\)
- For \(n = 7\): \(\{ \sqrt{7} \} \approx 2.646 \Rightarrow \{ \sqrt{7} \} = 3\)
- For \(n = 8\): \(\{ \sqrt{8} \} \approx 2.828 \Rightarrow \{ \sqrt{8} \} = 3\)
- For \(n = 9\): \(\{ \sqrt{9} \} = \{ 3 \} = 3\)
- For \(n = 10\): \(\{ \sqrt{10} \} \approx 3.162 \Rightarrow \{ \sqrt{10} \} = 4\)

2. **Continue this for all \(n\) up to 100**:
- From \(n = 11\) to \(n = 15\), \(\{ \sqrt{n} \} = 4\).
- From \(n = 16\) to \(n = 24\), \(\{ \sqrt{n} \} = 5\).
- From \(n = 25\) to \(n = 35\), \(\{ \sqrt{n} \} = 6\).
- From \(n = 36\) to \(n = 48\), \(\{ \sqrt{n} \} = 7\).
- From \(n = 49\) to \(n = 63\), \(\{ \sqrt{n} \} = 8\).
- From \(n = 64\) to \(n = 80\), \(\{ \sqrt{n} \} = 9\).
- From \(n = 81\) to \(n = 99\), \(\{ \sqrt{n} \} = 10\).
- For \(n = 100\): \(\{ \sqrt{100} \} = \{ 10 \} = 10\).

3. **Count the occurrences of each integer**:
- \(1\) appears \(1\) time.
- \(2\) appears \(3\) times (for \(n = 2, 3, 4\)).
- \(3\) appears \(5\) times (for \(n = 5, 6, 7, 8, 9\)).
- \(4\) appears \(7\) times (for \(n = 10, 11, 12, 13, 14, 15\)).
- \(5\) appears \(9\) times (for \(n = 16, 17, \ldots, 24\)).
- \(6\) appears \(11\) times (for \(n = 25, 26, \ldots, 35\)).
- \(7\) appears \(13\) times (for \(n = 36, 37, \ldots, 48\)).
- \(8\) appears \(15\) times (for \(n = 49, 50, \ldots, 63\)).
- \(9\) appears \(17\) times (for \(n = 64, 65, \ldots, 80\)).
- \(10\) appears \(19\) times (for \(n = 81, 82, \ldots, 100\)).

4. **Calculate the total sum**:
\[ \text{Total Sum} = 1 \times 1 + 2 \times 3 + 3 \times 5 + 4 \times 7 + 5 \times 9 + 6 \times 11 + 7 \times 13 + 8 \times 15 + 9 \times 17 + 10 \times 19 \]

- \(= 1 + 6 + 15 + 28 + 45 + 66 + 91 + 120 + 153 + 190\)
- \(= 715\)

### Final Answer:
The value of the series is \(715\).