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
- A. H. Beiler's Recreations in the Theory of Numbers, Dover, 1966
Number Curiosities
- Number 8 Is Interesting
- Curious Identities Involving Integer Squares
- Curious Identities Involving Integer Products
- Decimal Sums of Successive Integers
- Curious Identities In Pythagorean Triangles
- Hardy's Example of Non-Serious Theorems
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