TRIANGULAR NUMBERS First, he made pairs with all the numbers. So, the final answer that he got was 5050, which is the 100th triangular number. Next, he found out the total number of pairs he could make. Guass's teacher gave everyone a problem, which was to add up all the numbers from 1 to 100. (To find to 100th triangular number) She thought it would keep them occupied for a long time, but Guass immediately found the answer. Finally, he multiplied the sum of the 2 numbers in 1 pair by the number of pairs he could make in total. Triangular Numbers: Guass Ex. 1+100=101

2+99=101

3+98=101 ex. 100÷2=50.

He can make 50 pairs that add up to 101. ex. 101×50=5050 1 3 6 10 15 21 28 36 45 55 66 78 91 105 120 136 153 171 190 210 231 253 276 300 325 351 378 406 435 465 496 528 561 595 630 666 703 741 780 820 861 903 946 990 1035 1081 1128 1176 1225 1275 1326 1378 1431 1485 1540 1596 1653 1711 1770 1830 1891 1953 2016 2080 2145 2211 2278 2346 2415 2485 2556 2628 2701 2775 2850 2926 3003 3081 3160 3240 3321 3403 3486 3570 3655 3741 3828 3916 4005 4095 4186 4278 4371 4465 4560 4656 4753 4851 4950 5050 THE FIRST 100 TRIANGULAR NUMBERS ARE... The triangular numbers can be found in the third diagonal of Pascal`s triangle, starting at row 3 as shown in the diagram. The first triangular number is 1, the second is 3, the third is 6, the forth is 10, and so on. 1

1 1

1 2 1

1 3 3 1

1 4 6 4 1

1 5 10 10 5 1

1 6 15 20 15 6 1

1 7 21 35 35 21 7 1

1 8 28 56 70 56 28 8 1

1 9 36 84 126 126 84 36 9 1

FORMULA FOR FINDING TRIANGULAR NUMBERS n(n+1) 2 ex. what is the 64th triangular number?

64×(64+1)

64×65 = = 4, 160 4, 160 ÷ 2 = 2, 080 A triangular number is the number of dots in an equilateral triangle evenly filled with dots. For example, three dots can be arranged in a triangle; thus three is a triangle number. The nth triangular number is the number of dots in a triangle with n dots on a side. You get a triangular number by adding the next consecutive number to all the numbers before that. ex. 1+2+3+4+5=15 5th triangular number is 15 1 11 2 3 4 5 6 7 8 9 10 12 13 14 15 1st 2nd 3rd 4th 5th The sum of 2 consecutive triangular numbers equal a square number. 6 + 10 = 16 10 + 15 = 25 HANDSHAKE PROBLEM

Everyone in the room shakes hands with each other. (If you and I shake hands, that counts as one handshake only.) If there are 50 people, we would all shake hands...

1+2+3+4+5+6+........49 times The first person will make 49 new handshakes. Person #2 will make 49 handshakes, but one of those is the one he already made with Person #1. Person #3 makes 49 handshakes, but one is from Person #1 and another is from Person #2. So, each person makes one less new handshake than the person before. WORKSHEET THE END

QUESTIONS? BY MOLLY

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# Math-Triangular Numbers

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