Determine the sum of the first 20 terms of an A.P. if a4 - a2 = 4 and a1 + a3 + a5 + a6 = 30.

We'll write, for the beginning, the sum of n terms of an arithmetic series:

Sn = (a1+an)*n/2

a1 - the 1st term

an - the n-th term

n - the number of terms

Since n  = 20, we'll re-write the sum for the first 20 terms:

S20 = (a1 + a20)*20/2

S20 = (a1 + a20)*10

We'll have to calculate the first term and the common difference d, to determine any term of the arithmetic series.

From enunciation, we have:

a4 - a2 = 4

a4 = a1 + 3d

a2 = a1 + d

We'll write a4 and a2 with respect to a1 and d:

a1 + 3d - a1 - d = 4

We'll combine and eliminate like terms:

2d = 4

d = 2

 We also know, from enunciation, that:

a1 + a3 + a5 + a6 = 30

We'll write the terms with respect to a1 and d:

a1 + a1 + 2d + a1 + 4d + a1 + 5d = 30

We'll combine  like terms and substitute d:

4a1 + 11d = 30

4a1 = 30 - 11d

4a1 = 30 - 22

4a1 = 8

a1 = 2

Now, we can calculate a20:

a20 = a1 + 19d

a20 = 2 + 19*2

a20 = 2 + 38

a20 = 40

S20 = (a1 + a20)*10

S20 = (2 + 40)*10

S20 = 42*10

S20 = 420

The sum of the first 20 terms of the arithmetic progression is S20 = 420.