The third term of an arithmetical progression is 7, and the seventh term is 2 more than 3 times the third term. find the first term, the common difference and the sum of the first 20 terms.

General formula for n-th term of arithmetical progression is

a(n)=a(1)+d(n-1).

For 3d term we have

a(3)=a(1) +d(3-1),  where a(3)=7
7=a(1)+2d

For 7th term we have
a(7)=a(1) +d(7-1)
a(7)=a(1) + 6d

Also, we have that the seventh term is 2 more than 3 times the third term,
a(7)=3*a(3)+2= 3*7+2=21+2=23

So we have, a(7)=a(1) + 6d and a(7)=23. We can write
23=a(1) + 6d.

Now we can write a system of equations
23=a(1) + 6d
 - (7=a(1)+2d)
16 = 4d
d=4,

7=a(1)+2d

7=a(1)+2*4
a(1)=7-8=-1
a(1)= - 1

First term a(1)=-1, common difference d=4.

Sum of  the 20 first terms is
S=20 * (a(1)+a(20))/2

a(1)=-1
a(n)=a(1)+d(n-1)
a(20) = -1+4(20-1)=-1+4*19=75

S=20 * (-1+75)/2=74*10=740

Sum of 20 first terms is 740.