Sriman Dutta Posted August 22, 2016 Share Posted August 22, 2016 I observe a relationship here..... Binomial theorem depends on Pascal's triangle which goes as follows: 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 But, (11)^n also shows a similar sequence. Is there any relationship between these two ? Link to comment Share on other sites More sharing options...
fiveworlds Posted August 22, 2016 Share Posted August 22, 2016 Okay but after n=4 it fails completely ... Link to comment Share on other sites More sharing options...
Sriman Dutta Posted August 23, 2016 Author Share Posted August 23, 2016 Hmmm.....Right. Link to comment Share on other sites More sharing options...
Endy0816 Posted August 23, 2016 Share Posted August 23, 2016 Magic 11's is what you've happened upon. 1 5 10 10 5 1 -> 1 + 50 + 10*100 + 10*1000 + 5*10000 + 1*100000 = 161051 http://jwilson.coe.uga.edu/EMAT6680Su12/Berryman/6690/BerrymanK-Pascals/BerrymanK-Pascals.html 1 Link to comment Share on other sites More sharing options...
Country Boy Posted August 24, 2016 Share Posted August 24, 2016 Basically, this is because 11= 10+ 1. [math]11^2= (10+ 1)^2= 1(10^2)+ 2(10)(1)+ 1(1)= 100+ 20+ 1= 121[/math]. [math]11^3= (10+ 1)^3= 10^3+ 3(10^2)(1)+ 3(10)(1^2)+ 1^3= 100+ 300+ 30= 1= 1331[/math]. [math]11^4= 1(10^4)+ 4(10^3)(1)+ 6(10^2)(1^2)+ 6(10)(1^3)+ 1(1^4)= 10000+ 4000+ 600+ 40+ 1= 14641[/math]. it fails for n> 4 because the binomial coefficients then have more than 1 digit. Link to comment Share on other sites More sharing options...
DrKrettin Posted August 24, 2016 Share Posted August 24, 2016 That calculation looks as if it might be independent of the numbering system, so would be valid in, say, hexadecimal. If so, I guess it would work there for n=5 and until a coefficient had more than 1 digit in hex. Link to comment Share on other sites More sharing options...
Sriman Dutta Posted August 25, 2016 Author Share Posted August 25, 2016 Magic 11's is what you've happened upon. 1 5 10 10 5 1 -> 1 + 50 + 10*100 + 10*1000 + 5*10000 + 1*100000 = 161051 http://jwilson.coe.uga.edu/EMAT6680Su12/Berryman/6690/BerrymanK-Pascals/BerrymanK-Pascals.html Thanks Endy. It's really very helpful. I didn't knew that the Pascal's Triangle has so many interesting things. Link to comment Share on other sites More sharing options...
Endy0816 Posted August 26, 2016 Share Posted August 26, 2016 Thanks Endy. It's really very helpful. I didn't knew that the Pascal's Triangle has so many interesting things. Welcome. Reminded me of a similar pattern seen in Lychrel numbers. 196 + 691 = 7 18 7 (887) 887 + 788 = 15 16 15 (1675) 1675 + 5761 = 6 13 13 6 (7436) Link to comment Share on other sites More sharing options...
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