The arithmetic mean of the nine numbers in the given set $\{9, 99, 999, \dots, 999999999\}$ is a $9$-digit number $N$,all whose digits are distinct. The number $N$ does not contain the digit:

If $a, b, c$ are in Arithmetic Progression,then $(a - c)^2 = \dots$

Let there be four distinct integers in an increasing arithmetic progression. One of these integers is equal to the sum of the squares of the other three. What is the product of all these numbers?

If the sum of three numbers in an $A.P.$ is $24$ and their product is $440,$ find the numbers.

If $100$ times the $100^{th}$ term of an Arithmetic Progression with a non-zero common difference is equal to $50$ times its $50^{th}$ term,then what is its $150^{th}$ term?

Let $a, b, c, d, e$ be natural numbers in an arithmetic progression such that $a+b+c+d+e$ is the cube of an integer and $b+c+d$ is the square of an integer. The least possible value of the number of digits of $c$ is