Simplify the following using distributive property
$7 + 77 + 777 + 7 \times 9 + 77 \times 99 + 777 \times 999$ .

784770

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774770

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784780

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704770

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Solution

The correct option is D $784770$
$7 + 77 + 777 + 7 \times 9 + 77 \times 99 + 777 \times 999$
$= 7 (1 + 11 + 111 + 9 + 11 \times 99 + 111 \times 999)$ ,by distributive property
$= 7 (123) + 7 \times 9 (1 + 11 \times 11 + 111 \times 111)$
$= 7 (123) + 7 \times 9 (1 + 121 + 12321) = 7 (123) + 63 (12443) = 861 + 783909 = 784770$

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7 + 77 + 777 + 7 9 + 77 99 + 777 \times 999

7 + 77 + 777 + \dots \dots + (777 \dots 7 n

) =

\underset{n - digits}{. . . . . . . . . . .} 7 = \frac{7}{81} (10^{n + 1} - 9 n - 10)

Prove that $7 + 77 + 777 + . . . . . . + 777 . . . . . . . . . . . . . n - d i g i t s 7 = \frac{7}{81} (10^{n + 1} - 9 n - 10)$

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