Expert: Scott A Wilson - 9/20/2011

Question
QUESTION: I need to know if there is a function which describes the series below

The series is defined that an employee gets paid N for N days, then N+1 for the next N+1 days, and so on.

So he would get paid;

1,2,2,3,3,3,4,4,4,4,5,5,5,5,5

And I need the equation of his total return:

1,3,5,8,11,14,18,22,26,30,35,40,45,50,55

ANSWER: Note that in the series, there are 2-2's, 3-3's, 4-4's, etc.
This would make the pay the sum of squares before.
In other words, day 1 has a total of 1.

Days 2 and 3 have a total of 1 + 2(i-1), where i = 2 or 3.
That is the same as 2i - 1.

Days 4, 5, and 6 would have a total of 5 + 3(i-3), since 1+4=5.
That is the same as 3i - 4.

Days 7, 8, 9, and 10 would have a total of 14 + 4(i-6), since 1+4+9=14.
That is the same as 4i - 10.

Days 11, 12, 13, 14, and 15 would have a total of 30 + 5(i-10), since 1+4+9+16=30.
That is the same as 5i - 20.

Days 16, 17, 18, 19, 20, and 21 would have a total of 55 + 6(i-15).
That is the same as 6i - 35.

Every time the pay goes up by 1 per day to n units, the amount subtracted increases by n(n-1)/2.
This means that each time the amount paid is increased, the amount subtracted is the same as the summation from 1 to the current pay -1.

---------- FOLLOW-UP ----------

QUESTION: Scott,  thanks for your quick reply.  I might have misunderstood, but this doesn't answer my question. Im looking for the function where I get the total return for n days.  you method means I would have to iterate thru the series and know what the pay level is.

for example on day 5, with your method I would need to know that the pay is 3.

What I need is a function where the only input is the day number

ANSWER: The sum of 1 to n is n(n+1)/2.
To show this be examples,
 if n=1, n(n+1)/2 = 1*2/2 = 1;
 if n=2, n(n+1)/2 = 2*3/2 = 3; and
 if n=3, n(n+1)/2 = 3*4/2 = 6.

It can also be proved by taking 1+n + 2+(n-1) +  ....
If n is even, there are n/2 terms adding to n+1, so it is n(n+1)/2.
If n is odd, there are (n-1)/2 terms adding to n+1 and the last term in the middle is (n+1)/2.
So we get (n+1)(n-1)/2 + (n+1)/2 = (n^2-1)/2 + (n+1)/2 = (n^2+n)/2 = n(n+1)/2.

If the day is at the end of 2 years (730, or 731 with a leap year involved),
it can easily be found.  Take two times the number of days, which is 2*730 = 1460.
Using Excel, the squarerott(1460) = 38.20994635, try 38.
The sum of 1 through 38 is 38*39/2 = 19*39 = 741.
Since 741-38<730, 38 is the amount being made.


---------- FOLLOW-UP ----------

QUESTION: No, the answer for 3 days is 5, not 6.

Answer
The way to find the answer is to solve the equation k(k+1)/2 <= n for the greatest possible k.
Once this has been done, the answer is A + B where
A is [sum(m^2) for m going from 1 to k]   and
B is (k+1)P, where P is n - C, where C = [sum(m) from 1 to k].

For n = 1, k = 1.  This makes A be 1.  C is 1 as well, P = 1 - 1 = 0.  This makes B into 0.
Since the answer is A+B, then answer is 1.

At n=2, k is still 1, so A=1.  C = sum(m) from 1 to 1, so C=1, so n - C = 1.
Now k+1 = 2, so B = 2.  This makes A + B = 1 + 2 = 3, and that's the answer.

For n=3, k is 2 {since 2(2+1)/2=3 }, so A=1+4=5.  Now C = 1+2 = 3, so n-C = 0, so B=0,
so the answer, A+B, is 5.

For n=4, k is 2 again, so A is still 5.  Since k is still 2, C is still 3, so P = n-C = 1.
Since B = (k+1)P = 3*1 = 3, the answer is A+B = 5+3 = 8.

For n=5, k is still 2, so A is agaijn 5.  Since k is still 2, C is still 3, so P = n-C = 2.
Since B = (k+1)P = 3*2 = 6, the answer is A+B = 5+6 = 11.

For n=6, k is 3 { since 1+2+3=6 }, so A is 1+4+9 = 14.  Since k is 3, C is 1+2+3=6.
Since n-C=0, the answer is A, 14.

For n=7, k is still 3, so A is still 14.  Since C = 1+2+3 = 6, n-C = 1.
This makes B = (3+1)(1) = 4*1 = 4.  This makes the answer be 14+4 = 18.

I have worked it out in Excel; the formulas have been changed to what they are; in the spreadsheet, for example in row 7, n was A7, k was B7, A was C7, C was D7, P was E7, B was F7, and the answe was in G7.  There was also a * for multiplication.

The 1st column has n.
The 2nd column has k=TRUNC((-1+(1+8n)^0.5)/2).
The 3rd column has A=((2n+3)n+1)n/6.
The 4th column has C=k(k+1)/2.
The 5th column has P=n-C.
The 6th column has B=(k+1)P.
The 7th column has the answer, which is A+B.
n  k    A   C   P   B  Answer
1  1    1   1  0   0    1
2  1    1   1  1   2    3
3  2    5   3  0   0    5
4  2    5   3  1   3    8
5  2    5   3  2   6   11
6  3   14   6  0   0   14
7  3   14   6  1   4   18
8  3   14   6  2   8   22
9  3   14   6  3  12   26
10  4   30  10  0   0   30
11  4   30  10  1   5   35
12  4   30  10  2  10   40
13  4   30  10  3  15   45
14  4   30  10  4  20   50
15  5   55  15  0   0   55
16  5   55  15  1   6   61
17  5   55  15  2  12   67
18  5   55  15  3  18   73
19  5   55  15  4  24   79
20  5   55  15  5  30   85
21  6   91  21  0   0   91
22  6   91  21  1   7   98
23  6   91  21  2  14  105
24  6   91  21  3  21  112
25  6   91  21  4  28  119
26  6   91  21  5  35  126
27  6   91  21  6  42  133
28  7  140  28  0   0  140
29  7  140  28  1   8  148
30  7  140  28  2  16  156
31  7  140  28  3  24  164
32  7  140  28  4  32  172
33  7  140  28  5  40  180
34  7  140  28  6  48  188
35  7  140  28  7  56  196
36  8  204  36  0   0  204
37  8  204  36  1   9  213
38  8  204  36  2  18  222
39  8  204  36  3  27  231
40  8  204  36  4  36  240
41  8  204  36  5  45  249
42  8  204  36  6  54  258
43  8  204  36  7  63  267
44  8  204  36  8  72  276
45  9  285  45  0   0  285
46  9  285  45  1  10  295