Inequality that I believe can be conquered with Cauchy Schwarz I.E

Question

There are ten real numbers $x_0, . . . , x_9$ with $x_0 = 0, x_9 = 9$. What is the smallest possible value of the expression $$(x_1 − x_0)^2/1 +(x_2 − x_1)^2/2+(x_3 − x_2)^2/3+ · · · +(x_9 − x_8)^2/9$$? I thought this problem had cauchy schwartz I.E written all over it , I was able to get the $x_i$'s to telescope down to $9^2$ or $ 81$ but then Im getting confused what will be the bi in the sum. will it be the sum of $1+1/2+1/3...+1/9$ or just $1+2+3...+9$ ? please help :(

Answer 1

[![enter image description here][1]][1]

[1]: https://i.stack.imgur.com/fXUVO.jpg by cauchy schwarz the (xi+1 -x i ) telescope leaving 9^2=81 but then we have the denominators which are the sum of integers from 1 to 9 , this gives a total of 9/5

Answer 2

cauchy schwartz inequality with writing the denominators as a sequence of radicals ]1