Find an increasing arithmetic sequence of n real numbers whose sum is equal to N

Question

For example, i am looking for the sequence of 7 increasing numbers starting from 0 whose sum is 0.7 (more or less) with uniform distribution (the gap between the numbers are equal). It seems to me there is something to do with the pyramid but I did not find what i wanted into the web.

edit: this kind of sequence is usually called arithmetic progression

Answer 1

To summarize the discussion in the comments:

Given $n\in \mathbb N$ the OP is asking for a value $g_n$ such that the series $$0+g_n+2g_n+\cdots +(n-1)g_n=.7$$

We remark that this can be solved as $$.7=g_n\times (1+2+\cdots +(n-1))=g_n\times \frac {n(n-1)}2\implies \boxed {g_n=\frac {1.4}{n(n-1)}}$$

For instance, with $n=7$ we get $g_7=\frac 1{30}$.

Of course, there is nothing special about the value $.7$