Animated Solution for Mathematics - Sequence and Series: Let $S_n$ denote the sum of first $n$-terms of an arithmetic progression. If $S_{10}=530,S_5=140$, then $S_{20}-S_6$ is equal to :

Unlocking the Arithmetic Progression

Welcome, fellow traveler on the path to JEE mastery. Today, we are not just solving a problem; we are decoding the rhythmic heartbeat of an Arithmetic Progression (AP).

Imagine a sequence of numbers, a ladder where every step you take is of the exact same height. That height is our common difference, $d$, and the very first step we take is our starting term, $a$.

When we talk about $S_n$, the sum of the first $n$ terms, we are essentially measuring the total height of the first $n$ steps. Let us embark on this journey to find the hidden values of $a$ and $d$ and ultimately reach our destination: the value of $S_{20}-S_6$.

Decoding the DNA of the Sequence

Every AP is defined by its DNA: the first term $a$ and the common difference $d$. We are given two clues: $S_{10}=530$ and $S_5=140$.

Our master key is the elegant sum formula:

This formula is a bridge connecting the number of terms, the starting point, and the step size to the total sum. Let us apply this to our first clue. For $n=10$, we have:

Simplifying this, we get $5[2a+9d]=530$. Dividing both sides by $5$, we arrive at our first milestone, a clean linear equation:

Now, let us turn our attention to the second clue. For $n=5$, we have:

This simplifies to $\frac{5}{2}[2a+4d]=140$. Factoring out a $2$ from the bracket, we get $5[a+2d]=140$, which leads us to $a+2d=28$.

To make our algebraic dance easier, let us multiply this by $2$ to match the $2a$ term in our first equation:

The Algebraic Dance

We now have a system of two linear equations. This is where the magic happens.

By subtracting Equation 2 from Equation 1, we watch the $2a$ terms vanish into thin air:

With the common difference $d=10$ revealed, finding the first term $a$ is trivial. Substituting $d=10$ into Equation 2:

We have successfully decoded the DNA of our sequence: $a=8$ and $d=10$. We are now ready for the final act.

The Grand Finale

Our mission is to find $S_{20}-S_6$. We have the tools, we have the parameters, and we have the momentum.

Let us calculate $S_{20}$ first:

Next, we calculate $S_6$ with the same precision:

Finally, the moment of truth. We subtract the sum of the first six terms from the sum of the first twenty terms:

And there it is! The result is $1862$. You have navigated the system of equations, performed the calculations with care, and arrived at the correct answer.