Question 1209977
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For a positive integer n, let f(n) denote the integer that is closest to sqrt[4]{n}.  
Find the integer m so that sum_{n = 1}^m  f(n) = 100.
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        I solved this problem using MS Excel.

        My calculations are shown in the table below.



<pre>
First column of the table in the counter of natural numbers n = 1, 2, 3, . . . 

Second column is the values of {{{root(4,n)}}}, rounded to the closest integer number.

Third column is the sum S(n) of the first n integer numbers of the second column.


The table shows that the integer 'm' such that the sum S(m) is precisely 
equal to 100  is 48.


n            {{{root(4,n)}}}             S(n)
-----------------------------------------------    

1		1		1
2		1		2
3		1		3
4		1		4
5		1		5
6		2		7
7		2		9
8		2		11
9		2		13
10		2		15
11		2		17
12		2		19
13		2		21
14		2		23
15		2		25
16		2		27
17		2		29
18		2		31
19		2		33
20		2		35
21		2		37
22		2		39
23		2		41
24		2		43
25		2		45
26		2		47
27		2		49
28		2		51
29		2		53
30		2		55
31		2		57
32		2		59
33		2		61
34		2		63
35		2		65
36		2		67
37		2		69
38		2		71
39		2		73
40		3		76
41		3		79
42		3		82
43		3		85
44		3		88
45		3		91
46		3		94
47		3		97
48		3		100    <<<---===


So, the <U>ANSWER</U> to the problem's question is  m = 48.


Having this table, one can construct a wording solution, retelling this my solution in wording form
without using this table, but I prefer direct arguments.
</pre>

Solved.