If [x] is the greatest integer less than or equal to x, then find the value of the following series
`[sqrt1]+[sqrt2]+[sqrt3]+[sqrt4]+...+[sqrt(323)`

Options

A. 3237
B. 2373
C. 3723
D. none

Answer

To solve the problem of finding the value of the series \([ \sqrt{1} ] + [ \sqrt{2} ] + [ \sqrt{3} ] + [ \sqrt{4} ] + \ldots + [ \sqrt{323} ]\), we will follow these steps:

### Step 1: Understand the Greatest Integer Function
The greatest integer function \([x]\) gives the largest integer less than or equal to \(x\). For example:
- \([ \sqrt{1} ] = 1\)
- \([ \sqrt{2} ] = 1\)
- \([ \sqrt{3} ] = 1\)
- \([ \sqrt{4} ] = 2\)

### Step 2: Determine the Range of Values
We need to find the ranges for which \([ \sqrt{n} ]\) takes on specific integer values.
- For \(n = 1\) to \(3\), \([ \sqrt{n} ] = 1\) (3 terms)
- For \(n = 4\) to \(8\), \([ \sqrt{n} ] = 2\) (5 terms)
- For \(n = 9\) to \(15\), \([ \sqrt{n} ] = 3\) (7 terms)
- For \(n = 16\) to \(24\), \([ \sqrt{n} ] = 4\) (9 terms)
- For \(n = 25\) to \(35\), \([ \sqrt{n} ] = 5\) (11 terms)
- For \(n = 36\) to \(48\), \([ \sqrt{n} ] = 6\) (13 terms)
- For \(n = 49\) to \(63\), \([ \sqrt{n} ] = 7\) (15 terms)
- For \(n = 64\) to \(80\), \([ \sqrt{n} ] = 8\) (17 terms)
- For \(n = 81\) to \(99\), \([ \sqrt{n} ] = 9\) (19 terms)
- For \(n = 100\) to \(120\), \([ \sqrt{n} ] = 10\) (21 terms)
- For \(n = 121\) to \(143\), \([ \sqrt{n} ] = 11\) (23 terms)
- For \(n = 144\) to \(168\), \([ \sqrt{n} ] = 12\) (25 terms)
- For \(n = 169\) to \(195\), \([ \sqrt{n} ] = 13\) (27 terms)
- For \(n = 196\) to \(224\), \([ \sqrt{n} ] = 14\) (29 terms)
- For \(n = 225\) to \(255\), \([ \sqrt{n} ] = 15\) (31 terms)
- For \(n = 256\) to \(288\), \([ \sqrt{n} ] = 16\) (33 terms)
- For \(n = 289\) to \(323\), \([ \sqrt{n} ] = 17\) (35 terms)

### Step 3: Count the Contributions
Now we will count how many times each integer contributes to the sum:
- \(1\) appears \(3\) times
- \(2\) appears \(5\) times
- \(3\) appears \(7\) times
- \(4\) appears \(9\) times
- \(5\) appears \(11\) times
- \(6\) appears \(13\) times
- \(7\) appears \(15\) times
- \(8\) appears \(17\) times
- \(9\) appears \(19\) times
- \(10\) appears \(21\) times
- \(11\) appears \(23\) times
- \(12\) appears \(25\) times
- \(13\) appears \(27\) times
- \(14\) appears \(29\) times
- \(15\) appears \(31\) times
- \(16\) appears \(33\) times
- \(17\) appears \(35\) times

### Step 4: Calculate the Total Sum
Now we calculate the total contribution:
\[ \text{Total} = 1 \cdot 3 + 2 \cdot 5 + 3 \cdot 7 + 4 \cdot 9 + 5 \cdot 11 + 6 \cdot 13 + 7 \cdot 15 + 8 \cdot 17 + 9 \cdot 19 + 10 \cdot 21 + 11 \cdot 23 + 12 \cdot 25 + 13 \cdot 27 + 14 \cdot 29 + 15 \cdot 31 + 16 \cdot 33 + 17 \cdot 35 \]

Calculating each term:
- \(3 + 10 + 21 + 36 + 55 + 78 + 105 + 136 + 171 + 210 + 253 + 300 + 351 + 406 + 465 + 528 + 595 = 3723\)

### Final Answer
The value of the series \([ \sqrt{1} ] + [ \sqrt{2} ] + [ \sqrt{3} ] + [ \sqrt{4} ] + \ldots + [ \sqrt{323} ]\) is **3723**.