Q.The sums of $n$ terms of two A.P.s are in the ratio $(3n+8):(7n+15)$. Find the ratio of their 12th terms.

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:

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.

Write the same sum in reverse order:

Now add these two equations term by term. The first pair gives $a + [a+(n-1)d] = 2a + (n-1)d$. The second pair gives $(a+d) + [a+(n-2)d] = 2a + (n-1)d$. Every pair gives the same value $2a + (n-1)d$. There are $n$ such pairs.

Therefore:

This is the sum of the first $n$ terms of an A.P.

A More Natural Form

Notice that the last term of the A.P. is $l = a + (n-1)d$. So we can also write:

This is the average of the first and last terms, multiplied by the number of terms. That is exactly what the brick-pile intuition suggests: the average number of bricks per row is the average of the top and bottom rows, and the total is that average times the number of rows.

Example

Find the sum of the first 20 terms of the A.P.: $3, 7, 11, 15, \dots$ …