Animated Solution for Mathematics - Sequence and Series: Let $S_n$ denote the sum of first $n$ terms an arithmetic progression. If $S_{20}=790$ and $S_{10}=145$, then $S_{15}-S_5$ is :

Welcome, future engineers! Today, we are going to peel back the layers of an arithmetic progression. Many students look at a problem like this and see a wall of symbols. I want you to see a story.

An arithmetic progression is like a staircase where every step has a constant height, $d$. The first step is at height $a$. If you know $a$ and $d$, you know everything about that staircase. This is the 'DNA' of our sequence.

Decoding the Given Information

We are given two snapshots: the sum of the first twenty terms, $S_{20}=790$, and the sum of the first ten terms, $S_{10}=145$. Our master key is the sum formula:

This formula is not just a collection of variables; it is a map. It tells us that if we know the number of terms $n$, the starting point $a$, and the step size $d$, we can find the total sum.

Let's apply this to our first snapshot. For $n=20$, we have:

Simplifying this, we get $10[2a+19d]=790$, which reduces to the elegant linear equation:

Now, let's look at the second snapshot. For $n=10$, we have:

Simplifying this, we get $5[2a+9d]=145$, which reduces to:

The Power of Elimination

Now we have a system of two linear equations. This is where the magic happens. We have $2a+19d=79$ and $2a+9d=29$.

Notice how the $2a$ term is identical in both? This is a gift! If we subtract Equation (ii) from Equation (i), the $2a$ terms vanish into thin air.

We are left with $(19d-9d)=79-29$, which simplifies to $10d=50$. Thus, our common difference is $d=5$.

With $d=5$ in our pocket, finding $a$ is trivial. Substitute $d=5$ into Equation (ii):

Subtracting 45 from both sides gives $2a=-16$, meaning $a=-8$. We have successfully decoded the DNA of our sequence: $a=-8$ and $d=5$.

The Final Leap

Our target is to find $S_{15}-S_5$. Think of this visually. $S_{15}$ is the sum of the first 15 terms. $S_5$ is the sum of the first 5 terms.

When we subtract $S_5$ from $S_{15}$, we are removing the first 5 terms, leaving us with the sum of terms from $T_6$ to $T_{15}$.

First, let's calculate $S_{15}$:

Next, calculate $S_5$:

Finally, the difference:

There it is! The elegance of the result lies in the systematic reduction of the problem. You didn't just solve for a number; you navigated the structure of the sequence itself. Keep this mindset, and no problem will ever be too complex to solve. The final answer is 395.