If $m:n$ is the duplicate ratio of $m+x:n+x$, then $x^2=mn$.

Consider the following statements
I. $\underset{n\to \infty }{\lim }\frac{2^n+(-2{)}^n}{2^n}$ does not exist
II. $\underset{n\to \infty }{\lim }\frac{3^n+(-3{)}^n}{4^n}$ does not exist
Then

Three numbers are in the ratio 2:3:5. If the sum of their squares is 950, then what are the numbers?

If we divide 184 kg in the ratio 2 : 3 : 3, the relevant measures in kg are:

(A) 26, 79 and 79 

(B) 6, 89 and 89 

(C) 46, 79 and 79 

(D) 46, 69 and 69 

Of three positive numbers, the ratio of first and second is $5:6$ and that of second and third is $9:1$. The product of the first and third number is $480$. What is the sum of the three numbers? 

In a mixture of $165$ litres, the ratio of liquids $X$ and $Y$ is $6:5$. If $5$ litres of liquid $Y$ is added in the mixture, then what is the ratio of $X$ and $Y$ in the new mixture ?

If the sum of two numbers is equal to seven times of their difference, then what is the ratio of the numbers?

Let $x_k$ be real numbers such that $x_k\ge k^4+k^2+1$ for $1\le k\le 2018.$ Denote $N=\sum _{k=1}^{2018}k.$ Consider the following inequalities.
I. ${\left(\sum _{k=1}^{2018}k{x}_k\right)}^2\le N\left(\sum _{k=1}^{2018}k{x}_k^2\right)$

II. ${\left(\sum _{k=1}^{2018}k{x}_k\right)}^2\le N\left(\sum _{k=1}^{2018}{k}^2{x}_k^2\right)$
Then,