Answer:
The expression for the n-th term is [tex]a_n = -4+3n.[/tex]
Step-by-step explanation:
We know that the n-th term of an arithmetic progression is given by [tex]a_n= a+nd[/tex]. So we need to find the coefficients [tex]a[/tex] and [tex]d[/tex]. In order to do this, we substitute the values [tex]n=1[/tex] and [tex]n=2[/tex] in the expression for [tex]a_n[/tex]:
[tex]a_1 = a+d = -1,[/tex]
[tex]a_2 = a+2d = 2.[/tex]
Now, notice that we have two linear equations with two unknowns, which it is not difficult to solve. Thus, the solution is [tex]d=3[/tex] and [tex]a=-4[/tex].
Hence, the expression for the n-th term is [tex]a_n = -4+3n.[/tex]
We can check that the expression is correct substituting the values [tex]n=3[/tex] which gives [tex]a_3 = 5[/tex] and [tex]n=4[/tex] which gives [tex]a_4 = 8[/tex], as it is given in the statement of the problem.
