Note that

1111111=10^6+10^5+10^4+10^3+10^2+10+1

and

909091=10^6-10^5+10^4-10^3+10^2-10+1.

Compute the product of these two integers.

This is one way - I did it as the difference of 2 squares.

Heureka is doing it now as well, he probably has a better way...

$(10^6+10^5+10^4+10^3+10^2+10+1)\times (10^6-10^5+10^4-10^3+10^2-10+1)\\ = (10^6+10^4+10^2+1+10^5+10^3+10)\times (10^6+10^4+10^2+1-10^5-10^3-10)\\ =[ (10^6+10^4+10^2+1)+(10^5+10^3+10)]\times[ (10^6+10^4+10^2+1)-(10^5+10^3+10)]\\ =[ (10^6+10^4+10^2+1)^2-(10^5+10^3+10)^2]\\ =[ (10^6 (10^6+10^4+10^2+1)+ 10^4(10^6+10^4+10^2+1)+10^2 (10^6+10^4+10^2+1)+(10^6+10^4+10^2+1)] -(10^5+10^3+10)^2]\\ =[ 10^{12}+2*10^{10}+3*10^8+4*10^6+3*10^4+2*10^2+1]-[(10^5(10^5+10^3+10)+10^3(10^5+10^3+10)+10(10^5+10^3+10)]\\ =[ 10^{12}+2*10^{10}+3*10^8+4*10^6+3*10^4+2*10^2+1]\\ \qquad \quad-[10^{10}+2*10^8+3*10^6+2*10^4+10^2]\\ =[ 10^{12}+10^{10}+10^8+10^6+10^4+10^2+1]\\ =1010101010101$

1111111*909091=1,010,101,010,101= 239 * 4649 * 909091

Note that

1111111=10^6+10^5+10^4+10^3+10^2+10+1

and

909091=10^6-10^5+10^4-10^3+10^2-10+1.

Compute the product of these two integers.

geometric series:

$\begin{array}{|rcll|} \hline a_n &=& a_1 \cdot r^{n-1} \\ s_n &=& a_1 \left( \frac{-1+r^n}{1-r} \right) \qquad \text{sum}\\ \hline \end{array}$

geometric series 1:

$\begin{array}{|rcll|} \hline && 10^6+10^5+10^4+10^3+10^2+10+1 \qquad | \qquad a_1 = 1 \qquad r = 10 \\\\ s_n &=& a_1 \left( \frac{-1+r^n}{1-r} \right) \\ s_n &=& 1\cdot \left( \frac{-1+10^n}{1-10} \right) \\ s_7 &=& 1\cdot \left( \frac{-1+10^7}{1-10} \right) \\ s_7 &=&\left( \frac{-1+10^7}{1-10} \right) \\ \hline \end{array}$

geometric series 2:

$\begin{array}{|rcll|} \hline && 10^6-10^5+10^4-10^3+10^2-10+1 \qquad | \qquad a_1 = 1 \qquad r = -10 \\\\ S_n &=& a_1 \left( \frac{-1+r^n}{1-r} \right) \\ S_n &=& 1\cdot \left( \frac{-1+(-10)^n}{1-(-10)} \right) \\ S_7 &=& 1\cdot \left( \frac{-1+(-10)^7}{1+10} \right) \\ S_7 &=&\left( \frac{-1-10^7}{1+10} \right) \\ \hline \end{array}$

$\begin{array}{|rcll|} \hline && 1111111 \cdot 909091 \\ &=& (10^6+10^5+10^4+10^3+10^2+10+1) \cdot (10^6-10^5+10^4-10^3+10^2-10+1) \\ &=& s_7 \cdot S_7 \\ &=& \left( \frac{-1+10^7}{1-10} \right) \cdot \left( \frac{-1-10^7}{1+10} \right) \\ &=& \left[ \frac{(-1)^2-(10^7)^2}{1^2-(10)^2} \right] \\ &=& \frac{1-10^{14}}{1-10^2} \\ &=& \frac{10^{14}-1}{10^2-1} \\ &=& \frac{99999999999999}{99} \\ &=& 10101010101010 \\ \hline \end{array}$