The remainder when 1010 + 10100 + 101000 + . . . . . . + 101000000000 is divided by 7 is

Answer: Option B

Number of terms in the series = 10.
(We can get it easily by pointing the number of zeros in power of terms.
In 1^st term number of zero is 1, 2^nd term 2, and 3^rd term 3 and so on.)
$$\frac{{10}^{10}}{7},$$   Written as, $$\frac{{\left(7+3\right)}^{\left(4\times 2+2\right)}}{7}$$
The remainder will depend on $$\frac{3^2}{7}$$
So, remainder will be 2
$$\begin{array}{rl}& \frac{{10}^{1000}}{7},\text{remainder}=2\\ & \frac{{10}^{10000}}{7},\text{remainder}=1\end{array}$$
So, we get alternate 2 and 1 as remainder, five times each.
So, required remainder is given by
$$\begin{array}{rl}& \frac{\left(2+1+2+1+2+1+2+1+2+1\right)}{7}\\ & =\frac{15}{7}\end{array}$$
Remainder when 15 is divided by 7 = 1