If the first term of an A.P. is 3 and the sum of its first four terms is equal to one-fifth of the sum of the next four terms, then the sum of the first 20 terms is equal to

The first term of an A.P. is $$a = 3$$ and the sum of its first four terms equals one-fifth of the sum of the next four terms. We need to find $$S_{20}$$.

The sum of the first four terms is $$S_4 = \frac{4}{2}(2 \times 3 + 3d) = 2(6 + 3d) = 12 + 6d$$. The sum of the first eight terms is $$S_8 = \frac{8}{2}(2 \times 3 + 7d) = 4(6 + 7d) = 24 + 28d$$, and hence the sum of the fifth through eighth terms is $$S_8 - S_4 = 12 + 22d$$.

Using the given condition $$12 + 6d = \frac{1}{5}(12 + 22d)$$ and multiplying both sides by 5 gives $$60 + 30d = 12 + 22d$$, so $$8d = -48 \implies d = -6$$.

Substituting into the formula for the sum of the first twenty terms, we get $$S_{20} = \frac{20}{2}(2 \times 3 + 19(-6)) = 10(6 - 114) = 10(-108) = -1080$$, which is the correct answer Option 1: $$-1080$$.