fascinating triangular numbers

7/26/2019 Fascinating Triangular Numbers

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Fascinating Triangular Numbers

The numbers which can be arranged in a compact triangular pattern are termed as triangular numbers.The triangular numbers are formed by partial sum of the series 1+2+3+4+5+6+7......+n. o

T1! 1T2! 1 + 2 ! 3

T3! 1 + 2 + 3 ! 6T4! 1 + 2 + 3 + 4 ! 1"

o the nthtriangular number can be obtained as Tn! n#$n+1%&2' where n is any natural number.(n otherwords triangular numbers form the series 1'3'6'1"'15'21'2).....

*locs of birds often fly in this triangular formation. ,-en se-eral airplanes when flying togetherconstitute this formation. The properties of such numbers were first studied by ancient ree

mathematicians' particularly the /ythagoreans.

0a-e you heard of the following famous story about the famous mathematician arl *. auss.

The teacher ased e-eryone in the class to find the sum of all the numbers from 1 to 1"". To

e-erybodys surprise' auss stood up with the answer 5"5" immediately. The teacher ased him as to

how it was done. auss eplained that instead of adding all the numbers from 1 to 1""' add first andlast term i.e. 1 + 1"" !1"1' then add second and second last term i.e. 2 + !1"1 and so on. ,-erypair sum is 1"1 and their will be 5" such pairs $ total 1"" numbers in all to be added%' so 1"1 # 5" !

5"5" is the answer. o the sum of numbers from 1 to is $&2%#$+1%' where &2 are the number ofpairs and +1 is sum of each pair. This the famous formula for nthtriangular number.


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ome of the interesting properties of triangular numbers published in 58 are9

Curious properties of Triangular Numbers:

The sum of two consecuti-e triangular numbers is always a s:uare9

T1+ T2= 1 + 3 = 4 = 22

T2+ T3= 3 + 6 = 9 = 32

(f T is a Triangular number than #T + 1 is also a Triangular number9

9*T1+ 1 = 9 * 1 + 1 = 1 = T4 9*T2+ 1 = 9 * 3 + 1 = 2! = T"

; Triangular number can ne-er end in 2' 4' 7 or 9

(f T is a Triangular number than )#T + 1 is always a perfect s:uare9

!*T1+ 1 = ! * 1 + 1 = 9 = 32

!*T2+ 1 = ! * 3 + 1 = 2# = #2

The digital root $i.e. ultimate sum of digits until a single digit is obtained% of triangular

numbers is always 1'3'6 or .

The sum of n consecuti-e cubes starting from 1 is e:ual to the s:uare of n thtriangular number

i.e. Tn2! 13+ 23+ 33+ ... + n3

T42= 12= 1 = 13+ 23+ 33+ 43T#2= 1#2= 22# = 13+ 23+ 33+ 43+ #3

; triangular number greater than 1' can ne-er be a ube' a *ourth /ower or a *ifth power.

The sum of the s:uares of two consecuti-e triangular numbers is also a triangular number.

T12+ T22= 1 2+ 3 2= 1 = T4T22+ T32= 3 2+ 6 2= 4# = T9

T32

+ T42

= 6

2

+ 1

2

= 136 = T16 Tn$12+ Tn2= Tn2

$


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$;=B(C (=?B; from /atna' ?ihar submitted -ide his email dated 21


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There are pairs of triangular numbers such that the sum and difference of numbers in each pair

are also triangular numbers e.g. $15' 21%' $1"5' 171%' $37)' 7"3%' $7)"' "%' $14)5' 41)6%'$2145' 3741%' $546"' 67)6%' $7)75' )77)%... etc.9

21+ 1#= 36= T!: 21$ 1#= 6= T3

1"1+ 1#= 2"6= T23: 1"1$ 1#= 66= T11

"3+ 3"!= 1!1= T46: "3$ 3"!= 32#= T2#

and so on.

Some New Observations on Triangular Numbers :

There are some triangular numbers which are product of three consecuti-e numbers. *or

eample 12" is a triangular number which is product of three consecuti-e numbers 5' 6 and7.There are only 6 such triangular numbers' largest of which is 25)474216' as shown below9

1 * 2 * 3 = 6 = T3

4 * # * 6 = 12 = T1#

# * 6 * " = 21 = T2

9 * 1 * 11 = 99 = T44

#6 * #" * #! = 1!#136 = T6!

636 * 63" * 63! = 2#!4"4216 = T22"36

The triangular number 12" is the product of three' four and fi-e consecuti-e numbers.

4 * # * 6 = 2 * 3 * 4 * # = 1 * 2 * 3 * 4 * # = 12

o other triangular number is nown to be the product of four or more consecuti-e numbers.

There are infinite triangular numbers which are product of two consecuti-e numbers. *or

eample 6 is a triangular number which is product of two consecuti-e numbers 2 and 3. ome

others are as shown below9

2 * 3 = 6 = T3

14 * 1# = 21 = T2

!4 * !# = "14 = T119

492 * 493 = 242##6 = T696

2!" * 2!"1 = !239"" = T4#9


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16"3 * 16"31 = 2"99963 = T2366

9"#12 * 9"#13 = 9#!6!"6#6 = T13"93

#6!344 * #6!34# = 3231#4"6! = T!3"6

3312##4 * 3312### = 19"31"31#4" = T46!46#9

19369!2 * 19369!3 = 3"2"#9#"32##36 = T2"34196 etc

The triangular numbers which are product of two prime numbers can be termed as Triangular

emiprimes . *or eample 6 is a Triangular emiprimes . ome other eamples of Triangularemiprimes are9

1"'15'21'55'1'253'7"3'1")1'1711'1)1'27"1'34"3'5671'124"3'13)61'1531'1)721'25651'34453'3)5"3'4141'6""31'64261'73153'7""3')))31'1"4653'1"))11'

1144)1'126253'146611'15)2"3'1711'1))11'21)71'226)"1'25)121'26"11'2)6"3'351541'37153'3)5""3'3241'4)2653'475"3 etc as shown below9

2 * 3 = 6 = T3

3 * # = 1# = T#

3 * " = 21 = T6

# * 11 = ## = T1

" * 13 = 91 = T13

11 * 23 = 2#3 = T22

19 * 3" = "3 = T3"

@arsa5 Triangular Numbers:

@arsa5 -or Ni?en / numbers are tose numbers ;ic are 5i?isible b< teir sum of te5igits For eAample 1"29 - 19*91/ is 5i?isible b< 1+"+2+9 =19 so 1"29 is a @arsa5number

@arsa5 Triangular Numbercan be 5efine5 as te Triangular numbers ;ic are5i?isible b< te sum of teir 5igits For eAample Triangular number 112! is 5i?isible b curious pattern :

T1+ T2+ T3= T4

T#+ T6+ T"+ T!= T9+ T1

T11+ T12+ T13+ T14+ T1#= T16+ T1"+ T1!

T19+ T2+ T21+ T22+ T23+ T24= T2#+ T26+ T2"+ T2!

8ome i5entities :

Tn2= Tn+ Tn$1* Tn+1

Tn2$1= 2*Tn* Tn$1

For more on suc i5entities ?isit Terr< Trotter

Triangular numbers appear in %ascalDs Triangle n fact 3r5 5iagonal of %ascalDs Trianglegi?es all triangular numbers as so;n belo;:
http://www.trottermath.net/numthry/trident.htmlhttp://www.trottermath.net/numthry/trident.html


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11 1

12 11 3 31

1 4 64 11 # 1 1# 11 6 1#2 1#6 1

1 " 21 3# 3# 21" 11 ! 2!#6 " #6 2!! 1

1 9 36!4 126 126 !4 369 1

ne eAample of a %


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T1= T#+#= ## an5 so on

TT#+ TT6= T26

TT12+ TT14= T61

TT""+ TT!9= T3"6

TT1"4+ TT21= T!"1

TT1"9+ TT1249= T#396

TT243+ TT2!6= T121#1 an5 so on

@igl< Composite Triangular Numbers:

Numbers suc tat 5-n/ te number of 5i?isors of n is greater tan for an< smaller narecalle5 igl< composite numbers f n is a triangular number ten it can be terme5

as@igl< Composite Triangular Number For eAample 2! is a triangular number an55-2!/ = 6 Number of 5i?isors of all triangular numbers less tan 2! is less tan 6 8o 2!is a@igl< Composite Triangular number

>ll @igl< Composite Triangular numbers belo; #*113are:

1 3 6 2! 36 12 3 #2! 63 216 324 #46 2#2 "392 1#"! 43"#! "49"13!#2! 1493!#6 23112 216216 1"9"12 "6#"6# 136"2! 23621#9!!4216132 3996 4!192144 "#!91!16 "9663122 136"4#2!2366#644 491"23232 "!913224 12""46"2 ##""36444"2 66636#2"9#!!"6"!#2632 1"!"!3###14 242"462216 3"9!2"#94"2 242!"6#!#9#2 an526#"14631#!24

>bun5ant an5 'eficient Triangular Numbers:

Numbers suc tat s-n/ te sum of aliuot 5i?isors of n is greater tan nare calle5>bun5ant numbers f n is a triangular number ten it can be terme5 as >bun5antTriangular Number For eAample 36 is a triangular number an5 s-36/ = 1 + 2 + 3 + 4 + 6+ 9 + 12 + 1! = ## ;ic is greater tan 36 8o 36 is a >bun5ant Triangular number

>ll >bun5ant Triangular numbers belo; 1#are:

3666"!12212"633"!#2!63666"!!299112!11"613261#41#961""1!32162!23462##6262!2!#31632434!63#"3!2!49#42"!4#646#649##46#""!#!!66216632!6"!6"14"26"626"!"#!2#6!646!""!91!9"39!"12961441!"!1162!1291224612"212!!13#3142!1419614"61#41##"61611162916!361"21""661"9##1!3361!#2!1911199212"629121#2!21"362194#22#"!2322234362492431249"62#22#!"!261626#6#26"962"9662!6!2!922964629!9362!3!"6

3162631!"!3264336"339334"16349!3#""!3"12!3"9#3!226396393444"4132!41616424!64366439#644!#4#1#46#646364"#!64!#164!!2!49""#14#136#2326#26##362!#39#6#42!###2"!#62!#6616#"63#"9"#!996#93463"!6"266142#61""66212!63#466462649!666666436"!96696693"!"#"239"3#36"392"#"!"6636""2!"!21"!66"9!!2


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!1!1!221#!32!!3436!4666!632!6"36!"99!!41!96"69191!6939693#2!94!396#!9"29!346 an5 9!"9

Numbers suc tat s-n/ te sum of aliuot 5i?isors of n is less tan nare calle5 'eficientnumbers f n is a triangular number ten it can be terme5 as 'eficient TriangularNumber For eAample 21 is a triangular number an5 s-21/ = 1 + 3 + " = 11 ;ic is lesstan 21 8o 21 is a 'eficient Triangular number

>ll 'eficient Triangular numbers belo; 1#are:

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

!#"!!#491!#9#!"1#3!"#"1!!!31!92#39#2#99#1913"!9223#9266#939619439#9#2669#"3961419"4619"939923# an5 996!1

Curious 8um of T;o Triangular Numbers

Ta&e an eAample of a 1$5igit Triangular number 161444!3# t can be seen tat tistriangular number is te sum of te 1614tan5 44!3#ttriangular numbers 8o te sumof t;o triangular numbers is eual to te number forme5 from concatenation of in5eA oftese t;o triangular numbers

T1614+ T44!3#= 161444!3#

ter eAamples are:

T9+ T41#= 941#

T#!#+ T91= #!#91

T12+ T1#4#= 121#4#

T1#+ T1"26= 1#1"26

T244+ T2196= 2442196

T"+ T36"6= "36"6


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T"69+ T3!46= "693!46

T14"4+ T#226= 14"4#226

T2!29+ T69"= 2!2969"

T33+ T"1"1= 33"1"1 an5 so on

Can


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Hit a= an5 b= te follo;ing recursi?e euations can gi?e furter ?alues of a an5b:

an=3 * an$1+ 2 * bn$1+ 2

bn=4 * an$1+ 3 * bn$1+ 3

-submitte5 b< r o< Ilatcfor5 ?i5e is email 5ate5 1" ar 211/

Can an


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4 'u5ene: >ca5emic %ress199#

11 Trotter T Jr 8ome 5entities for te Triangular Numbers. Recr Math6 12!$13#19"3 ttp:;;;trottermatnetnumtr

fascinating triangular numbers

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