1952 AHSME Problems/Problem 50
Problem
A line initially 1 inch long grows according to the following law, where the first term is the initial length.
\[1+\frac{1}{4}\sqrt{2}+\frac{1}{4}+\frac{1}{16}\sqrt{2}+\frac{1}{16}+\frac{1}{64}\sqrt{2}+\frac{1}{64}+\cdots\] (Error making remote request. Unexpected URL sent back)
If the growth process continues forever, the limit of the length of the line is:
Solution
We can rewrite our sum as the sum of two infinite geometric sequences.
![\[1 + \frac{1}{4}\sqrt{2} + \frac{1}{4} + \frac{1}{16}\sqrt{2} + \frac{1}{16} + ...\]](http://latex.artofproblemsolving.com/c/b/a/cbac2ec7ad58e5c966aeec70c5eefabc2e766447.png)
![\[=\]](http://latex.artofproblemsolving.com/b/7/f/b7f0dbf30cb7ab74a726ffc9b15d33af79a3267c.png)
![\[(1 + \frac{1}{4} + \frac{1}{16} + \frac{1}{64} + ...) + (\frac{1}{4}\sqrt{2} + \frac{1}{16}\sqrt{2} + \frac{1}{64}\sqrt{2} + ...)\]](http://latex.artofproblemsolving.com/3/9/b/39ba564284fead5f022dbc0b50411414dcd0b368.png)
![\[(1 + \frac{1}{4} + \frac{1}{16} + \frac{1}{64} + ...) + \sqrt{2}(\frac{1}{4} + \frac{1}{16} + \frac{1}{64} + ...)\]](http://latex.artofproblemsolving.com/f/5/3/f5358e5bbcff092cee0ea98f5dbf2de0562d2f7b.png)
We now take the sum of each of the infinite geometric sequences separately
See also
| 1952 AHSC (Problems • Answer Key • Resources) | ||
| Preceded by Problem 49 |
Followed by Last Question | |
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