Animated Solution for Mathematics - Sequence and Series: If $\alpha ,\beta$ are natural numbers such that $100^2-199\beta =(100)(100)+(99)(101)+(98)(102)+\cdots +(1)(199)$, then the slope of the line passing through $(\alpha ,\beta )$ and origin is :

Imagine you are standing before a long, complex-looking series. It is easy to feel overwhelmed by the sheer number of terms, but as a mathematician, your job is to look past the noise and find the underlying structure.

The series given is $S=(100)(100)+(99)(101)+(98)(102)+\cdots +(1)(199)$.

At first glance, it is just a collection of products. However, the first number in each pair is decreasing, while the second is increasing. This pattern is the heartbeat of the problem.

The Key to the Kingdom

To master this series, we must define its general term, $T_r$. By observing the pattern, we can express the $r$-th term as $T_r=(100-r)(100+r)$, where $r$ ranges from $0$ to $99$.

This is where the magic happens. We are not just adding numbers; we are manipulating algebraic forms. By applying the difference of squares identity, $(a-b)(a+b)=a^2-b^2$, our general term simplifies beautifully:

Suddenly, the complexity vanishes, replaced by a simple subtraction.

Now that we have our simplified general term, we can express the entire sum $S$ using Sigma notation:

Because Sigma is a linear operator, we can split this into two distinct parts:

The first part is a constant sum. We are adding $100^2$ to itself $100$ times (from $r=0$ to $99$), which gives us $100\times 100^2=100^3$.

The second part is the classic sum of squares, using the formula ${\sum }_{k=1}^n{k}^2=\frac{n(n+1)(2n+1)}{6}$. Substituting $n=99$, we get:

Combining these, we find $S=100^3-199\times 1650$. Comparing this to the form $100^{\alpha }-199\beta$, we identify $\alpha =3$ and $\beta =1650$.

The final step is to find the slope of the line passing through $(3,1650)$ and the origin $(0,0)$. The slope $m$ is calculated as:

Even when a problem contains a typo, your ability to derive the truth through logic is what defines you as a JEE aspirant. You have navigated the complexity, simplified the expression, and arrived at the truth. The final result is 550.