E115 Un impair, deux pairs, trois impairs..
Solution de Nicole Guinamard
soit suite a ... .1, 2 4, 5 7 9, 10 12 14 16, 17 19 21 23 25...
soit suite b = le rang de chaque élément...1, 2 3, 4 5 6, 7 8 9 10 , 11 12 13 14 15 ...
On observe :
suite b-(a-b)=..1.../....2...2.../...3...3...3.../...4...4...4...4.../...5...5...5...5...5.../
suite ...(a-b)=..o.../....o...1.../...1...2...3.../...3...4...5...6../...6...7...8...9...10../
suite
....b...=...1.../...2...3.../...4...5...6.../...7...8...9...10.../...11...12...13...14...15..
...=...1...+2...+3...+4...+5..../
suite B...=...(1+2)=1*3=3./(1+2)+3=2*3=6../.(1+4)+(2+3)=2*5=10 /..(1+4) +(2+3)+5 = 3*5 = 15 suite....
a...=...1.../...2...4.../...5...7...9.../....10....12...14...16.../....17...19...21...23....25../
...= (1²)./.(2²-2)(2²)./.(3²-4)(3²-2)(3²)./.(4²-6)(4²-4)(4²-2)(4²)./.(5²-8)(5²-6)(5²-4)(5²-2)(5²)../
suite A...=..1²../...2²../...3²../...4²../...5²
suite b-(a-b)=...6...6...6...6...6...6.../....7...7...7...7...7...7...7..../...
suite....(a-b)=...10...11...12...13...14...15../....15...16...17...18...19...20...21.../...
suite ....b...=...16...17...18...19....20....21../....22...23...24...25...26...27....28../...
...=...+6.../...+7.../...
suite B...=...(1+6)+(2+5)+(3+4)= 3*7 = 21../...= 4*7 =28.../...
...(6/2)*(6+1)/...[(7+1)/2]*7/...
suite.... a...=...26...28...30...32...34...36../...37...39...41...43...45...47...49.../...
...=(6²-10)(6²-8)(6²-6)(6²-4)(6²-2)(6²)/.(7²-12)(7²-10)(7²-8)(7²-6)(7²-4)(7²-2)(7²)../...
suite A ...=...6²../...7²../...
suite b-(a-b)=....8...8...8...8...8...8...8...8../...9...
suite....(a-b)=....21...22...23...24...25...26...27...28../..28...
suite ....b...=...29...30...31...32...33...34...35....36../..37...
...=...+8.../...
suite B ...=...= 4*9 = 36../...
...(8/2)*(8+1)/...
suite.... a...=...50...52...54...56...58...60...62....64../..65...
...=(8²-14)(8²-12)(8²-10)(8²-8)(8²-6)(8²-4)(8²-2)(8²)./(9²-16)...
suite A...=...8²../...
su b-(a-b)=...62../...63...63...63...63...63...63...63...63..../...64...
suite(a-b)=1891../...1946...1947...1948....1949...1950...1951....1952...1953../..1953...
suite
..b..=1953../...2009...2010...2011....2012...2013....2014...2015..2016../...2017...
...=..+62../...+63../...
suite B..= 31*63/...32*63../...
...(62/2)*(62+1)...[(63+1)/2]*63...
suite..a.=3844../..3845 ...3955...3957 ...3959 ...3961...3963...3965...3967...3969 ../..3970...
...62²/(63²-124)... (63²-14)(63²-12)(63²-10)(63²-8)(63²-6)(63²-4)(63²-2)(63²)/(64²-126)..
suite A = ...62².../...63²../...
Voilà est-ce que mon « tableau » parle assez ? Sauf erreur , 3955, serait donc le 2009ème élément J'ai trouvé 31*63..<2009<32*63 en tâtonnant un peu, ce qui n'est pas très glorieux ...
mais amusant quand même et fort joli ma foi !