Q.The sum of three numbers in A.P. is 24 and their product is 440. Find the numbers.

Sum of First $n$ Terms of an Arithmetic Progression

Imagine you are stacking bricks in a triangular pile. The bottom row has 10 bricks, the next row has 9, then 8, and so on, until the top row has 1 brick. How many bricks are there in total? You could count them one by one, but there is a much smarter way.

The Intuition: Pairing Terms

An Arithmetic Progression (A.P.) is a sequence where each term differs from the previous one by a fixed number, called the common difference ($d$). For example: $2, 5, 8, 11, 14, \dots$ has $d = 3$.

Now, suppose we want the sum of the first $n$ terms of an A.P. Let the first term be $a$ and the common difference be $d$. The terms are:

$a, \ a+d, \ a+2d, \ a+3d, \ \dots, \ a+(n-1)d$

The trick is to write the sum forward and then backward, and add them.

Let $S_n$ be the sum of the first $n$ terms.

$S_n = a + (a+d) + (a+2d) + \dots + [a+(n-1)d]$ …