Animated Solution for Mathematics - Sequence and Series: Let $S_n$ denote the sum of the first n terms of an A.P. If $S_4=16$ and $S_6=-48$, then $S_{10}$ is equal to :

Welcome, future engineer. Today, we are not just solving a math problem; we are decoding the DNA of a sequence.

An Arithmetic Progression (A.P.) is a sequence where the difference between consecutive terms is constant. It is a rhythm, a heartbeat of numbers.

When we are given the sum of the first $n$ terms, $S_n$, we are essentially given the 'total energy' of that sequence up to a certain point. Our mission is to find the sum of the first ten terms, $S_{10}$, using only the clues provided by $S_4$ and $S_6$.

The Sum Formula

Every journey begins with the right tool. For an A.P., the sum of the first $n$ terms is governed by the elegant formula:

Here, $a$ is the first term, and $d$ is the common difference. Think of $a$ as your starting position and $d$ as your velocity.

If you know these two, you know everything about the sequence. The problem gives us two snapshots: $S_4=16$ and $S_6=-48$.

These are our two equations. We have two unknowns, $a$ and $d$. This is a classic system of linear equations waiting to be solved.

Let us translate the first clue, $S_4=16$, into math. Substituting $n=4$ into our formula, we get:

Simplifying this, $2(2a+3d)=16$, which reduces beautifully to $2a+3d=8$. Let's call this Equation 1.

Now, for the second clue, $S_6=-48$. Substituting $n=6$:

This simplifies to $3(2a+5d)=-48$, or $2a+5d=-16$. This is Equation 2.

Now, look at these two equations side-by-side. They are perfectly aligned. By subtracting Equation 1 from Equation 2, the $2a$ terms vanish, leaving us with a simple equation for $d$:

We have found the common difference! It is negative, which tells us our sequence is decreasing rapidly.

Now, finding $a$ is just a matter of substitution. Plugging $d=-12$ into $2a+3d=8$ gives $2a-36=8$, so $2a=44$, meaning $a=22$.

We have our parameters: $a=22$ and $d=-12$. We are ready to find $S_{10}$.

We plug these values back into the sum formula one last time:

And there it is. The sum of the first ten terms is $-320$.

It is a result of careful, step-by-step logic. Remember, in the JEE, the complexity isn't in the math itself, but in your ability to remain calm and methodical when the numbers start to get negative. You have mastered the sequence.