Arithmetic sequence with too many known values

[Guest]

I need to split a value into decreasing amounts where I know the start and end values and the number of steps to do it in.
How can I calculate this? I have tried applying Arithmetic Sequence formulae, but because I know so many fixed values they don't work. Any ideas?

My example data is:
Start value D=5
Number of steps N=6
First step S1=1
Last step Sn=0.5

How can I calculate the size of the 4 steps that fit between the first and last step?

[Math Tutor]

I cannot understand the problem.

This is not clear:

Start value D=5
Number of steps N=6
First step S1=1
Last step Sn=0.5

[Guest]

Sorry, maybe I was too brief.

Imagine a line that is 5mm long I want to split that line up into 6 segments.
I know that the first segment must be 1mm and the last segment must be 0.5mm.
The segments between the first and last segments must decrease in size in a linear fashion.
How do I work out the size of the 4 segments between the first and last segment?

Re-phrasing the example data in relation to the expanded description:
Line length L=5
Number of segments N=6
First segment S1=1
Last segment S6=0.5

How do I work out S2,S3,S4,S5?

[Math Tutor]

Here are arithmetic sequence formulas.
[tex]a_n = a_1 + (n - 1)d[/tex]
[tex]a_n = 0.5[/tex]
[tex]n=6[/tex]
[tex]a_1 = 1[/tex]

then:
[tex]d = \frac{a_n - a_1}{n - 1}[/tex]
[tex]d = \frac{0.5}{5} = 0.1[/tex]
[tex]a_2 = a_1 + d = 1.1[/tex]
[tex]a_3 = a_1 + 2*d = 1.2[/tex]

Do you understand the solution?

[Guest]

I can see what you are suggesting and this is basically what I have been doing so far.
However, in your example you are adding [tex]d[/tex]:
[tex]a_2=a_1+d=1.1[/tex]
but my sequence starts at 1.0 and decreases to 0.5 so I think [tex]d[/tex] should be subtracted like this:
[tex]a_2=a_1-d=0.9[/tex]

Completing this sequence:
[tex]d=\frac{0.5}{5}=0.1[/tex]
[tex]a_1=1.0[/tex]
[tex]a_2=a_1-d=0.9[/tex]
[tex]a_3=a_1-2*d=0.8[/tex]
[tex]a_4=a_1-3*d=0.7[/tex]
[tex]a_5=a_1-4*d=0.6[/tex]
[tex]a_6=a_1-5*d=0.5=a_n[/tex]

But if I then add the resulting values they don't add up to the start value. If you recall my line length [tex]L[/tex] was 5.0, but adding the resulting sequence values results in 4.5:
[tex]a_1+a_2+a_3+a_4+a_5+a_6=1+0.9+0.8+0.7+0.6+0.5=4.5[/tex]

The problem is that I want all the values to add up to the total line length.

[Math Tutor]

Yes you are right d = -0.1

[Math Tutor]

But it is not possible
the sum of the members to be 5 and

Number of segments N=6
First segment S1=1
Last segment S6=0.5

If this is true it is is not an arithmetic progression.