Sum of the first n terms of an AP is:

To find the sum of the first $n$ terms of an Arithmetic Progression (AP), let the terms be: $$a, a+d, a+2d, \dots, a+(n-1)d$$ Let $S_n$ represent the sum of these $n$ terms: $$S_n = a + (a+d) + (a+2d) + \dots + [a+(n-1)d]$$ If we write the series in reverse order: $$S_n = [a+(n-1)d] + [a+(n-2)d] + \dots + a$$ Adding these two equations vertically, term by term, each pair results in the same value: $$2S_n = [2a + (n-1)d] + [2a + (n-1)d] + \dots + [2a + (n-1)d]$$ Since there are $n$ terms in the series: $$2S_n = n \times [2a + (n-1)d]$$ Dividing both sides by 2, we get the standard formula: $$S_n = \frac{n}{2} [2a + (n-1)d]$$ Thus, the correct option is B.