Geometric Progression
Write out a series of three or more different whole numbers,
starting from 1, so that the numbers add up to a square.
Thus, 1+2+4+8+16+32=63, but this is just one short of being a
square.
I am only aware of two answers in whole numbers, and these
are easy to discover. I am very interested in any other solutions
that might be found, as well as any algebraic solutions.
Source: Henry Ernest Dudeny, Curious Problems & Puzzles #243
Ken's Solutions:
The two I know of are:
1 + 3 + 9 + 27 + 81 = 121 = 11^2
1 + 7 + 49 + 343 = 400 = 20^2
Interestingly, I ran a quick simulation to check for solutions
up to (1e8)^2 and found no other solutions!
Feel free to find others though, and let me know if you do!
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