I try to distribute beams on an horizontal array, with increasing spacings according to this series.
1+1=2 2+2=4 4+3=7 7+4=11 11+5=16 16+6=22
I'd like to feel what it looks like and learn how to change the "rate of growth" it has.
I tried to put: xn = n(n-1)/2 + 1 (found here) into this online graph builder.
Didn't work, welcome any help. I'm also intersted in any kind of "slow growing function"
The formula for the sum of a general quadratic sequence is:
$$S_2(n,s,d_1,c) = \cfrac{n(cn^2 + 2c + 3nd_1 + 6s - 3cn - 3d_1)}{6}$$
Where $n$ is the number of terms to be summed, $s$ is the starting term of the series, $d_1$ is the first difference (subtracting the first term from the second term) and $c$ is the constant difference between the differences.
From your series $1 + 2 + 4 + 7 + \ ...$ we can see that $s = 1$, $d_1 = 1$, $c = 1$.
So you want the $n$th term here, which means we can calculate the sum up to and including $n$ terms then subtract the sum up to and including $n-1$ terms. Plugging in the values it gives us the $n$th term (denoted $T_n$):
$$T_n = S_2(n,1,1,1) - S_2(n-1,1,1,1) = \frac{n(n-1)}{2} + 1$$
So it is correct, don't know why it didn't show it as correct for you though.