The fourth term of an A.P. is equal to three times the first term and the seventh term exceeds twice the third term by one .Calculate the common...

We'll note the first term as a1 and the common difference
as d.

Now, we'll put in relations the constraints from
enunciation:

"The fourth term of an A.P. is equal to three
times the first term":

a4  =
3a1

But a4 = a1 + 3d

a1 + 3d =
3a1

We'll combine like
terms:

2a1 = 3d

We'll divide
by 2 both sides:

a1 =
3d/2

 The second constraint from enunciation is:"the
seventh term exceeds twice the third term by one"

a7 = 2a3
+ 1

But a7 = a1 + 6d

a1 + 6d =
2a3 + 1, where a3 = a1 + 2d

a1 + 6d = 2a1 + 4d +
1

We'll combine like terms:

a1
= 2d-1

But a1 = 3d/2

3d/2 
=2d - 1

3d = 4d -
2

d =
2

a1 = 3d/2 

a1
= 3*2/2

a1 =
3