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.