Physics problem: torque of forces

Question

I came across this task on a website and have been curious about its solution, but I’m having trouble figuring it out:

First, let's assume that the springs barely stretch.

A long, rigid pallet of negligible mass is suspended on "n" equally spaced springs with the same directional force and the same unstretched length. The first spring can exert a maximum force of "D", the second 2×D, .., the nth n×D without breaking. What is the maximum weight of a body that can be placed on the pallet? Where should we put it?

Answer 1

It seems to be that the maximum weight is achieved when all springs are working to support the weight (I prefer pillars to springs as we shouldn't have to deal with the springs compressing differentially)

The maximum weight should be summation from k = 1 to n of kd or n(n-1)D/2

The weight should be located at the center of mass so that net torque is 0

Summation from k = 1 to n of ((k-1)L/(n-1) - x_COM)*kD where the term in parentheses is the lever arm for each force and kD the magnitude of each force.

We can split um the summatiion

LD/(n-1) sum k² - LD/(n-1)sum k - x_COMW = 0

LD/n-1)(n*(n+1)(2n+1)/6 - n(n+1)/2) - x_COMn(n+1)D/2 = 0

x_COM = L*2/3 !!! I can't help thinking there was a better way....

I tried it for n=6 and it worked. (remember spacing of springs will be 6/5)

Thanks for the problem. Take care.