Animated Solution for Mathematics - Binomial Theorem: The coefficient of $x^{101}$ in the expression $(5+x{)}^{500}+x(5+x{)}^{499}+x^2(5+x{)}^{498}+\cdots +x^{500}$, $x>0$, is

The given series is:

At first glance, this appears to be a complex algebraic expression. However, by observing the relationship between consecutive terms, we can identify the underlying structure.

If we divide the second term $x(5+x{)}^{499}$ by the first term $(5+x{)}^{500}$, we obtain the ratio $\frac{x}{5+x}$. Similarly, dividing the third term by the second yields the same ratio.

This confirms the series is a Geometric Progression (G.P.) with: First term $a=(5+x{)}^{500}$ Common ratio $r=\frac{x}{5+x}$ * Number of terms $n=501$ (since the power of $x$ ranges from $0$ to $500$)

We utilize the sum formula for a G.P., $S=a\frac{1-r^n}{1-r}$. Substituting our values:

The denominator simplifies as follows:

Substituting this back into the expression for $S$:

By simplifying the fractions and canceling the $(5+x{)}^{501}$ terms, we arrive at the elegant result:

We are tasked with finding the coefficient of $x^{101}$ in $S$. We rewrite the expression as:

The term $-\frac{x^{501}}{5}$ does not contribute to the coefficient of $x^{101}$. Therefore, we focus solely on the expansion of $\frac{(5+x{)}^{501}}{5}$.

Using the Binomial Theorem, the general term of $(5+x{)}^{501}$ is given by:

To find the coefficient of $x^{101}$, we set $r=101$:

Finally, we account for the divisor of $5$ present in our expression for $S$:

Applying the laws of exponents, $\frac{5^{400}}{5^1}=5^{399}$. Thus, the final coefficient is: