How can i answer this progression question?

Question

the 8th term of an arithmetic progression is 5 times the 4th term.If the second term is -3, find

(a) the common difference

(b)the first term

Answer 1

For an arithmetic progression, the nth term, a_n , is given by the formula:

 

   a_n = a₁ + (n-1)d, where d is the common difference.

 

So, we have a₂ = a₁ + d = -3

 

           and a₁ + 7d = 5(a₁ + 3d)

 

Simplifying, we get the system of equations:  a₁ + d = -3

                                                               4a₁ +8d = 0

 

From the second equation, a₁ = -2d

 

Since a₁ = -2d and a₁ + d = -3, we get -2d + d = -3

 

 Therefore,  d =3 and  a₁ = -2d = -6