First slide

Geometric progression

Question

$(666 \dots 6)^{2} + (888 \dots 8)$ is equal to
n digits n digits

Moderate

A

$\frac{4}{9} (10^{n} - 1)$
B

$\frac{4}{9} (10^{2 n} - 1)$
C

$\frac{4}{9} {(10^{n} - 1)}^{2}$
D

None of these

Solution

$(666 \dots 6) = 6 + 6 \times 10 + 6 \times 10^{2} + \dots + 6 \times 10^{n - 1}$
n digits
$\begin{array}{l} = 6 (1 + 10 + 10^{2} + \dots + 10^{n - 1}) \\ = \frac{6}{9} (10^{n} - 1) = \frac{2}{3} (10^{n} - 1) \end{array}$
Similarly, $(888 \dots 8) = \frac{8}{9} (10^{n} - 1)$
n digits
Hence, required sum
$\begin{array}{l} = \frac{4}{9} {(10^{n} - 1)}^{2} + \frac{8}{9} (10^{n} - 1) \\ = \frac{4}{9} (10^{2 n} - 2 \cdot 10^{n} + 1 + 2 \cdot 10^{n} - 2) \\ = \frac{4}{9} (10^{2 n} - 1) \end{array}$

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