Let's begin by listing out the information given to us:
1st term = 359, 2nd term = 352, 3rd term = 345
[tex]\begin{gathered} 359,352,345\ldots x_n \\ x_1=359,x_2=352,x_3=345 \\ x_1-x_2=x_2-x_3\Rightarrow359-352=352-345\Rightarrow7=7 \\ 7=7 \end{gathered}[/tex]This is an Arithmetic Progression (A.P.)
[tex]\begin{gathered} x_1=359 \\ x_2=359-7(2-1)\Rightarrow359-7(1)=359-7=352 \\ x_3=359-7(3-1)\Rightarrow359-7(2)=359-14=345 \\ x_n=x_1-7(n-1) \\ n_{38}=x_1-7(38-1)=359-7(37)=359-259=100 \\ n_{38}=100 \end{gathered}[/tex]