Difference between revisions of "PaperMath’s sum"
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Papermath’s sum states, | Papermath’s sum states, | ||
− | <math>\sum_{i=0}^{2n-1} {( | + | <math>\sum_{i=0}^{2n-1} {(10^ix^2)}=(\sum_{j=0}^{n-1}{(10^j3x)})^2 + \sum_{k=0}^{n-1} {(10^k2x^2)}</math> |
Or | Or | ||
− | <math>x^2\sum_{i=0}^{2n} {10^i}=( | + | <math>x^2\sum_{i=0}^{2n-1} {10^i}=(3x \sum_{j=0}^{n-1} {(10^j)})^2 + 2x^2\sum_{k=0}^{n-1} {(10^k)}</math> |
For all real values of <math>x</math>, this equation holds true for all nonnegative values of <math>n</math>. When <math>x=1</math>, this reduces to | For all real values of <math>x</math>, this equation holds true for all nonnegative values of <math>n</math>. When <math>x=1</math>, this reduces to | ||
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==Proof== | ==Proof== | ||
− | First, note that the <math>x^2</math> part is trivial multiplication | + | First, note that the <math>x^2</math> part is trivial multiplication, associativity, commutativity, and distributivity over addition, |
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+ | Observing that | ||
+ | <math>\sum_{i=0}^{n-1} {10^i} = | ||
+ | (10^{n}-1)/9</math> | ||
+ | and | ||
<math>(10^{2n}-1)/9 = 9((10^{n}-1)/9)^2 + 2(10^n -1)/9</math> | <math>(10^{2n}-1)/9 = 9((10^{n}-1)/9)^2 + 2(10^n -1)/9</math> | ||
− | + | concludes the proof. | |
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==Problems== | ==Problems== | ||
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==Notes== | ==Notes== | ||
− | Papermath’s sum was named by the aops user Papermath. The name is not widely used. | + | Papermath’s sum was named by the aops user Papermath, after noticing it in a solution to an AMC 12 problem. The name is not widely used. |
==See also== | ==See also== | ||
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*[[Cyclic sum]] | *[[Cyclic sum]] | ||
*[[Summation]] | *[[Summation]] |
Latest revision as of 17:07, 13 October 2024
Contents
PaperMath’s sum
Papermath’s sum states,
Or
For all real values of , this equation holds true for all nonnegative values of . When , this reduces to
Proof
First, note that the part is trivial multiplication, associativity, commutativity, and distributivity over addition,
Observing that and concludes the proof.
Problems
AMC 12A Problem 25
For a positive integer and nonzero digits , , and , let be the -digit integer each of whose digits is equal to ; let be the -digit integer each of whose digits is equal to , and let be the -digit (not -digit) integer each of whose digits is equal to . What is the greatest possible value of for which there are at least two values of such that ?
Notes
Papermath’s sum was named by the aops user Papermath, after noticing it in a solution to an AMC 12 problem. The name is not widely used.