SEMINAR - UČENJE ISTRAŽIVANJEM I RJEŠAVANJEM
PROBLEMA
Željka MILIN ŠIPUŠ
Ljiljana ARAMBAŠIĆ
Matija BAŠIĆ
ISTRAŽIVANJE
“MNOGOKUTNI BROJEVI”:
Trokutni brojevi: 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, …
Kvadratni brojevi: 1, 4, 9, 16, 25, …
Peterokutni brojevi: 1, 5, 12, 22, …
Dokaz indukcijom!
1
( 1)
2
n n
n
T T n
n nT
2
(4, ) (4, 1) (2 1)
(4, )
P n P n n
P n n
(5, ) (5, 1) (3 2)
(3 1)(5, )
2
P n P n n
n nP n
ISTRAŽIVANJE
“MNOGOKUTNI BROJEVI”
Navedite formule za opće članove
❖ trokutnih brojeva:
❖ kvadratnih brojeva
❖ peterokutnih brojeva
❖ šesterokutnih brojeva
❖ …
❖ m-terekutnih brojeva
2
( 1)3
2
(2 0)4 (4, )
2
(3 1)5 (5, )
2
(4 2)6 (6, ) (2 1)
2
(5 3)7 (7, )
2
(6 4)8 (8, ) (3 2)
2
n
n nT
n nP n n
n nP n
n nP n n n
n nP n
n nP n n n
( , )P m n
OPĆA FORMULA
Dokaz indukcijom!
ISTRAŽIVANJE
( , ) (( 2) ( 4))2
...
( 1)( 2)
2
nm P m n m n m
n nm n
“VEZE”
1. Svaki je mnogokutni broj povezan s trokutnim:
a)
b)
2. Rekurzivna veza kod mnogokutnih m-brojeva
3. Svaki je šesterokutni broj ujedno i trokutni
ISTRAŽIVANJE
1 1( , ) ( 2) ( 3)n n nP m n m T n m T T
1( 1, ) ( , ) nP m n P m n T
1 2 1
( 1) 2 (2 1)(6, ) 4 4
2 2n n
n n n nP n T n n T
( , 1) ( , ) ( 2) 1P m n P m n m n
KORIJEN IZ 3 I TROKUTNI BROJEVI
MATHOLOGER – MATEMATIČKI VIDEO
Burkard Polster, Njemačka, afilijacija: Monash University, Melbourne, Australia
”Visualising irrationality with triangular squares”
https://youtu.be/yk6wbvNPZW0
1. Uočite autorovu majicu!
2. Uočite rečenicu “This is incredibly beautiful proof!” (do 2.5min)
KORIJEN IZ 3 I TROKUTNI BROJEVI
MATHOLOGER – MATEMATIČKI VIDEO
Burkard Polster, ”Visualising irrationality with triangular squares”
QUESTIONS:
1. What are “triangular squares”?
2. Claim: Square root of 3 is irrational. Prove it with algebra!
3. Claim: “Nearest miss solutions” are good approximations of square root of 3. How do we form them?
4. What are triangular triangles?
5. Gauss’s claim: Every positive integer is a sum of at most three triangular numbers.
“A smaller equation follows from the larger one, so why doesn’t it prove, just as in the case of triangular squares, that three identical triangular numbers cannot add to another triangular number?”
MOŽE LI VRIJEDITI SLJEDEĆA TVRDNJA?
NOVA TEMA – PITALICA
0
2 1n
n
GEOMETRIJSKI NIZ
PONOVIMO
0
11
2nn
1 1 1, , ,...
2 4 8
PONOVIMO
REDOVI
1. Definicija reda
2. Kad kažemo da red konvergira?
3. Odredite sumu konvergentnog geometrijskog reda! Uz koje uvjete vrijedi ta formula?
4. Koje se još situacije mogu dogoditi osim konvergencije?
5. Vrijedi li
0
1( 1) ?
2
n
n
DAKLE…
DAKLE…
MATHOLOGER – MATEMATIČKI VIDEO
“9.999... really is equal to 10”
https://www.youtube.com/watch?v=SDtFBSjNmm0
MOGUĆE TEME ZA SEMINAR
MATHOLOGER – MATEMATIČKI VIDEO
Burkard Polster, ”Visualising irrationality with triangular squares”
QUESTIONS:
1. “Nearest miss solutions” are good approximations of square root of 3. How do we form them?
2. What are triangular triangles?
3. Gauss’s claim: Every positive integer is a sum of at most three triangular numbers.
“A smaller equation follows from the larger one, so why doesn’t it prove, just as in the case of triangular squares, that three identical triangular numbers cannot add to another triangular number?”
MOGUĆE TEME ZA SEMINAR
1. Znate li još neke konvergentne redove osim geometrijskih?
2. Konvergira li red
Kolika mu je suma?
3. Konvergira li red
Kolika mu je suma? (Baselski problem)
1 1 1 11 ...?
2 3 4 5
1 1 1 11 ...?
4 9 16 25
MOGUĆE TEME ZA SEMINAR
1. Znate li još neke konvergentne redove osim geometrijskih?
2. Konvergira li red
Kolika mu je suma? Ln (2)
3. Konvergira li red
Kolika mu je suma? (Baselski problem) π2/6
1 1 1 11 ...?
2 3 4 5
1 1 1 11 ...?
4 9 16 25
MOGUĆE TEME ZA SEMINAR
REDOVI KOJI NE KONVERGIRAJU
Srinivasa Ramanujan (Indija, 1887- 1920)
11 2 3 4 5 ...
12
MOGUĆE TEME ZA SEMINAR
REDOVI KOJI NE KONVERGIRAJU
MATHOLOGER
Numberphile v. Math: the truth about 1+2+3+...= 1/12
https://www.youtube.com/watch?v=YuIIjLr6vUA
Riemann's Rearrangement Theorem
Riemann's paradox: pi = infnity minus infinity
https://www.youtube.com/watch?v=-EtHF5ND3_s
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