Question

Study the given pattern and find the rule. Then, using the rule, write the next five steps. 1+1=11 11 x 11 = 121 111 x 111 =12321 1111 x 1111 =1234321​

Answer 1

Answer:

$11 \times 11 = 121 \\ 111 \times 111 = 12321 \\ 1111 \times 1111 = 1234321 \\ 11111 \times 11111 = 123454321 \\ 111111 \times 111111 = 12345654321 \\ 1111111 \times 1111111 = 1234567654321 \\ 11111111 \times 11111111 = 123456787654321 \\ 111111111 \times 111111111 = 12345678987654321$

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Answer 2

Answer:

Because 111111111 times 111111111 equals 12345678987654321 according to the laws of multiplication for real numbers.

If your question is more along the lines of “Why is there such a nice pattern in this product,” then perhaps I can elaborate a bit further.

Note that 111111111 = 100000000 + 10000000 + … + 100 + 10 + 1. As a consequence of the distributive property of multiplication over addition,

111111111×111111111=111111111×(100000000+10000000+⋯+100+10+1)=11111111100000000+1111111110000000+⋯+11111111100+1111111110+111111111.

Writing this out vertically, we get

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Note that the first column has one 1, the second column has two 1s, and so on, with the middle column having nine 1s, the next column having eight 1s, etc.

Therefore, the sum is 12345678987654321.