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