Note that the terms of an arithmetic series is given by the formula:

a_n = a_1 + (n – 1)d

where a_n is the current term, a_1 is the first term, d is the common difference.

Also note, that the sum of an arithmetic sequence is given by:

S_n = n(a_1 + a_n)/2 = n[2a_1 + (n – 1)d]/2

Now we have the common difference d = 2a_1, and the sum of the first 6 terms, S_6 = 6(a_1 + a_6)/2 = a_1^2

=> a_n = a_1 + (n – 1)2a_1 = a_1 + 2na_1 – 2a_1 = (2n – 1)a_1

=> a_6 = 11a_1

Now S_6 = a_1^2

=> 6(a_1 + a_6)/2 = a_1^2

=> 2a_1^2 = 6a_1 + 6a_6

=> 2a_1^2 = 6a_1 + 6(11a_1)

=> 2a_1^2 = 6a_1 + 66a_1

=> 2a_1^2 = 72a_1 …………………this is a quadratic equation in a_1, you can replace a_1 with a variable, say x, if all the symbols are getting you confused.

=> 2a_1^2 – 72a_1 = 0

=>2a_1(a_1 – 36) = 0

Since a_1 cannot be zero,

a_1 – 36 = 0

=> a_1 = 36

So the first term of the sequence is 36, and if you're interested, the sequence isa_n = 36 + (n – 1)72

That is 36, 108, 180, 252, 324, 396,…