osman@sprite.DEC (Eric Osman, dtn 283-7484) (02/12/85)
I originally sent this to RHEA::DECWRL::"net.math,net.puzzles". I got
an error back saying net.puzzles was wrong, so now I'm sending it to
just net.puzzle.
Newsgroups: net.math,net.puzzles
To: net.math, net.puzzles
Subject: solve 1 + 11 + 111 . . . + n 1's
Let n1 be the last number in the sequence, i.e. n 1's.
The answer for a given n is 1 + 11 + 111 + . . . + n1
a(n) = (10*n1 - n) / 9
The way I solved it was to observe that the sum is a portion of an infinite
converging geometric series, so I started with the series and shaved off
everything but our desired sum.
The useful geometric series, illustrated for n = 4 is
1111.111... + 111.111... + 11.111... + 1.111... + .111... +
.0111... + .00111... + . . .
The first number in the series, 1111.1111... (please continue to allow my loose
flipping between example and generality) we can represent as the largest
term, L, so we have
L = n1 + 1/9
If we call the entire geometric series g, we quickly derive value of g,
like this:
First,
g = L + L*10^-1 + L*10^-2 + L*10^-3 . . .
Multiply both sides by 10:
10 * g = L*10 + L + L*10^-1 + L*10^-2 + . . .
Note that g appears in second equation, so we can write it as
10 * g = L*10 + g
so finally we complete the rederivation of geometric series summation and
have
g = 10 * L / 9
The first useful thing to strip off are all the 0.111...'s. There are
exactly n+1 of them. That's a constant we can call c.
c = (n+1) * .111... = (n+1) / 9
Next, we want to strip off the residue of all the other non-integer terms
of g, which will finally leave us with what we are looking for. So
we'll strip off
r = .0111... + .00111... + .000111... . . .
This is itself a geometric series, computable the same way we did with g,
giving:
r = 1/81.
The answer we seek of 1 + 11 + 111 + 1111 + . . . + n1 is
a(n) = g - c - r
Simplifying, this is merely
a(n) = (10*n1 - n) / 9
Try it for 4. 1 + 11 + 111 + 1111 = 1234. Does it check ? . . .
a(4) = (10*1111 - 4) / 9 = (11110 - 4) / 9 = 11106 / 9 = 1234
Yippee !!
At this point, I smiled and went to bed.