Decimal Sums of Successive Integers

What comes next in the following sequence?

\(1 \times 9 + 2 = 11\)

\(12 \times 9 + 3 = 111\)

\(123 \times 9 + 4 = 1111\)

\(1234 \times 9 + 5 = 11111\)

\(12345 \times 9 + 6 = 111111\)

\(123456 \times 9 + 7 = 1111111\)

\(1234567 \times 9 + 8 = 11111111\)

\(12345678 \times 9 + 9 = 111111111\)

\(123456789 \times 9 + 10 = 1111111111\)

A road to answer to this question lies in the distinction between arithmetic and algebra - specific and general [Beiler, 57-58]. The trick is to express all nine of the above identites as specific cases of a more general one. We can do that! All nine appear to be instances of

\((10 - 1) \times (1\cdot 10^{n-1}+2\cdot 10^{n-2} + \ldots + r\cdot 10^{n-r}+\ldots + n) + (n+1)\).

Multiply out and collect the terms with the same powers of 10 (these will obviously be in the form \((r+1)-r=1\):

\(10^{n}+10^{n-1} + \ldots + 1 = \frac{10^{n+1}-1}{10-1}=111\ldots 1\),

with \(n+1\) successive \(1\)'s.

So, we may now add entries to the above table:

\(1234567900 \times 9 + 11 = 11111111111\)

\(12345679011 \times 9 + 12 = 111111111111\)

\(123456790122 \times 9 + 13 = 1111111111111\)

\(1234567901233 \times 9 + 14 = 11111111111111\)

\(12345679012344 \times 9 + 15 = 111111111111111\)

\(123456790123455 \times 9 + 16 = 1111111111111111\)

\(1234567901234566 \times 9 + 17 = 11111111111111111\)

etc.

This may not be as visually appealing as the shorter, original table. As a payoff, there is a great satisfaction of knowing the mechanics behind what - at first sight - might have appeared as a (math) mystery.

Yet, there are similar wonders. For example:

\(9 \times 9 + 7 = 88\)

\(98 \times 9 + 6 = 888\)

\(987 \times 9 + 5 = 8888\)

\(9876 \times 9 + 4 = 88888\)

\(98765 \times 9 + 3 = 888888\)

\(987654 \times 9 + 2 = 8888888\)

\(9876543 \times 9 + 1 = 88888888\)

\(98765432 \times 9 + 0 = 888888888\)

and also

\(1 \times 8 + 1 = 9\)

\(12 \times 8 + 2 = 98\)

\(123 \times 8 + 3 = 987\)

\(1234 \times 8 + 4 = 9876\)

\(12345 \times 8 + 5 = 98765\)

\(123456 \times 8 + 6 = 987654\)

\(1234567 \times 8 + 7 = 9876543\)

\(12345678 \times 8 + 8 = 98765432\)

\(123456789 \times 8 + 9 = 987654321\)

References

  1. A. H. Beiler's Recreations in the Theory of Numbers, Dover, 1966

Number Curiosities

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