Animated Solution for Mathematics - Binomial Theorem: The remainder on dividing $1+3+3^2+3^3+\cdots +3^{2021}$ by 50 is ____.

The problem asks us to find the remainder when the sum $S=1+3+3^2+3^3+\cdots +3^{2021}$ is divided by $50$.

This expression is a Geometric Progression (G.P.) where the first term $a=1$ and the common ratio $r=3$.

To determine the number of terms $n$, we observe the exponents range from $0$ to $2021$. Thus, the total number of terms is:

Using the sum formula for a G.P., $S_n=\frac{a(r^n-1)}{r-1}$, we substitute our known values:

We now need to evaluate $S(\mathrm{mod}50)$. To facilitate this, we manipulate the base $3$ to relate it to the divisor $50$.

We note that $3^2=9$, which can be written as $(10-1)$. Thus, we rewrite the expression as:

We apply the Binomial Theorem to expand $(10-1{)}^{1011}$:

All terms containing $10^k$ where $k\ge 2$ are multiples of $100$. We can group these as $100\lambda$ for some integer $\lambda$.

The remaining terms are:

Substituting this expansion back into our expression for $S$:

Simplifying the fraction, we obtain:

To find the remainder when $S$ is divided by $50$, we look at $5054(\mathrm{mod}50)$:

Since $5000$ and $50$ are both divisible by $50$, the remainder is $4$.

The final remainder is 4.