Animated Solution for Mathematics - Sequence and Series: If 19th term of a non-zero A.P. is zero, then its (49th term) : (29th term) is:

Welcome, future engineer! Today, we are going to peel back the layers of a seemingly simple problem. In the world of JEE Advanced, we often encounter questions that look like basic algebra but are actually tests of your ability to see the underlying structure of a sequence.

Let's dive into the beauty of the Arithmetic Progression (A.P.).

Defining the Sequence

Imagine you are standing on a number line. An A.P. is like taking consistent, equal-sized jumps. We define our starting position as $a$ and the size of each jump as $d$.

The $n$-th term, $T_n$, is simply where you land after $n-1$ jumps from your starting point. Mathematically, we write this as:

This formula is the heartbeat of all linear sequences. It tells us that the value of any term is just the initial position plus the accumulated displacement caused by the common difference.

We are told that the 19th term is zero. This is our anchor! When we plug this into our formula, we get:

This simplifies beautifully to $a+18d=0$, or $a=-18d$.

Think about what this means: the first term is exactly 18 steps behind the origin. By expressing $a$ in terms of $d$, we have reduced our two-variable problem into a single-variable world. We are no longer guessing; we are calculating.

Now, let's find the 49th term. Using our formula, $T_{49}=a+48d$. Substituting our anchor value $a=-18d$, we get:

Similarly, for the 29th term, $T_{29}=a+28d$. Substituting again:

Do you see the symmetry emerging? We have transformed the 49th and 29th terms into simple multiples of $d$.

The Ratio

Finally, we are asked for the ratio $T_{49}:T_{29}$. When we set up the fraction, the common difference $d$—which we didn't even know the value of—simply vanishes:

The final result is 3.

A Faster Way

If you want to think like a topper, look at the 'Smart Visual Method'. Since $T_{19}=0$, we can treat the 19th term as our origin.

The 29th term is $29-19=10$ steps away from the origin, so $T_{29}=10d$. The 49th term is $49-19=30$ steps away, so $T_{49}=30d$.

The ratio is simply:

This is the power of understanding the nature of the sequence rather than just blindly following the algebra. You've mastered the logic, and in doing so, you've mastered the problem. Keep this intuition sharp—it will serve you well in the exam hall!