Question 1:   Can numbers $1, \sqrt{2}, \sqrt{3}$ be the terms of an arithmetic progression?

Question 2:   Find the first term and the difference of the arithmetic progression  ($a_n$ )  if $a_5+a_{12}=41; a_{10}+a_{14}=62$ .

Question 3:   What is the value of $x$ when expressions $x^2-4,5x+3,3x+2$ in this right order form an arithmetic progression?

Question 4:   In which case the equation $a_1\times a_4=a_2^2$ works for the terms of an arithmetic sequence?

Question 5:   In what order should expressions ${\left(a+b\right)}^2,{\left(a-b\right)}^2,a^2+b^2$ be put to form an arithmetic sequence?

Question 6:   The sum of the first $75$ terms of an arithmetic progression ($a_n$ ) equals $450$ . Find $a_{38}$ .

Question 7:   Can the sequence with the sum of the first  $n$  terms $S_n=a\times n^2+b\times n \left(a\in R, b\in R\right)$ , be an arithmetic progression?

Question 8:   Find the sum of all the natural numbers that are multiple of $4$ and less than $130$ .

Question 9:   The initial term of an arithmetic sequence  ($a_n$ )  is $a_1=-9$ and its difference $d=6$.  How many first terms of the progression make the sum $S=960$ ?

Question 10:   Which five sequent numbers of an arithmetic progression  ($a_n$ )  $3,7,11\dots$ make the sum $S=135$ ?