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First row:          1

Second row:   2 , 3

Third Row:    4 , 5 , 6

Fourth row: 7 , 8 , 9 , 10

 

What is the sum of the numbers in row 2019?

 

-tommarvoloriddle

EDIT

What I tried:

 

Using Formulas (like finding the terms one and the arithmetic sequences one... my teacher said it was wrong... lol...)

eating grapes

 

Why I need you to show work:

 

So I can understand, both the problem and the solution.

 

WARNING:

only answer the question if you are fine with getting badgered by a confused person(me).

 Jul 11, 2019
edited by tommarvoloriddle  Jul 11, 2019
edited by tommarvoloriddle  Jul 11, 2019
 #1
avatar
+3

This pyramid has the following "closed form":

1/2[n^3 + n], where n = row number

1/2[2019^3 + 2019] =4,115,087,439

 Jul 11, 2019
edited by Guest  Jul 11, 2019
edited by Guest  Jul 11, 2019
 #7
avatar+1713 
+1

Thank you so much!

tommarvoloriddle  Jul 11, 2019
 #10
avatar+1713 
+1

But why to the third power? can you explain? plssssssssssssss

tommarvoloriddle  Jul 11, 2019
 #2
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+5

Look   at   the last number in each row  and notice this pattern

               1     =     sum of the 1st positive integer

                  3   =    sum of the first two positive integers

                     6 =    sum of the first three positive integers

                        10  =  sum of the first 4 positive integers

 

So.....the last entry in the nth row apears to be just the sum of the 1st  n positive integers

 

And the sum of the first n positive integers is just   :  n (n + 1)   /2

 

So....the last entry  in  the 2019th row is the sum of the first 2019 positive integers  = 

 

(2019) (2020)  / 2  =   2039190

 

And the nth row contains n integers...so....

 

We will have 2019  integers in this row and we need to find the first one....

 

Note that, after the first row....the first entry on any row is given by  :

 

Last entry  - row number + 1 

 

So...the first entry on the 2019th row must be :  2039190 - 2019 + 1  = 2037172

 

So...finally....the sum of the terms in the 2019th row is given by

 

[ first entry + last entry ] * number of terms / 2  =

 

[2037172 + 2039190 ] * [ 2019] / 2  = 

 

4,115,087,439

 

 

cool cool cool

 Jul 11, 2019
 #6
avatar+1713 
+1

Thx so much!

tommarvoloriddle  Jul 11, 2019
 #3
avatar+130493 
+5

To generate a "closed" form solution that the Guest found......, we can use something known as the " sum of differences"

 

We have the following  sequence of row sums

 

1      5       15     34      65       111    take the positive difference between terms until we get a "constant" row

    4      10     19     31        46

         6       9      12     15

              3      3       3

 

It took us three  rows of differences to get to a "constant" row

This means that we will have a cubic (3rd power) "generating" polynomial  of the form

 

an^3  + bn^2  + cn + d = S(n)         where n is the row  number and S(n) is the row sum

 

So...we have this system of  equations 

a(1)^3 + b(1)^2 + c(1) + d  = 1

a(2)^3 + b(2)^2 + c(2) + d = 5

a(3)^3 + b(3)^2 + c(3) + d  = 15

a(4)^3  + b(4)^2 + c(4) + d  = 34   which gives us

 

a  +  b  +  c  + d   =   1

8a  + 4b + 2c + d  = 5

27a + 9b + 3c + d  = 15

64a + 16b + 4c + d  = 34

 

I'll  spare you the gory details of solving this by hand  [ but...I can go through it...if you want to ]

 

The solutions  are   a = 1/2, b = 0, c = 1/2   d = 0

 

So....the polynomial that generates the sum of any nth row is given by

 

(1/2)n^3 + (1/2)n  =  S(n)

 

[ n^3 +  n ]

________    =   S(n)

      2

 

Just as the "Guest" found!!!

 

 

cool cool cool

 Jul 11, 2019
edited by CPhill  Jul 11, 2019
 #4
avatar+1713 
+1

Thx!

tommarvoloriddle  Jul 11, 2019
 #5
avatar+1713 
+1

Thx! so much

tommarvoloriddle  Jul 11, 2019
edited by tommarvoloriddle  Jul 11, 2019
 #8
avatar+26398 
+5

What is the sum of the numbers in row 2019?

 

First row:1Second row:2,3Third Row:4,5,6Fourth row:7,8,9,10

 

 

rowsum11=122,3(2+32)2=534,5,6(4+62)3=1547,8,9,10(7+102)4=34511,12,13,14,15(11+152)5=652019a,,b(a+b2)2019= ?nan,,bn(an+bn2)n=sn

 

an= ?
First row:d1=124711Second Row:d2=1234Third row:d3=111

an=(n10)d1+(n11)d2+(n22)d3=(n10)1+(n11)1+(n22)1=1+(n1)1+(n2)(n1)211=1+n1+(n2)(n1)2an=n+(n2)(n1)2

 

bn= ?

First row:d1=1361015Second Row:d2=2345Third row:d3=111

bn=(n10)d1+(n11)d2+(n22)d3=(n10)1+(n11)2+(n22)1=1+(n1)2+(n2)(n1)211=1+2n2+(n2)(n1)2bn=2n1+(n2)(n1)2

 

 

sn=(an+bn2)n=(n+(n2)(n1)2+2n1+(n2)(n1)22)n=(3n1+(n2)(n1)2)n=(3n1+n23n+22)nsn=(n2+12)n

 

The sum of the numbers in row 2019

sn=(n2+12)ns2019=(20192+12)2019s2019=20381812019s2019=4115087439

 

laugh

 Jul 11, 2019
edited by heureka  Jul 11, 2019
 #9
avatar+1713 
+6

Thank you- You really didn't have to multiply that out, did you?

tommarvoloriddle  Jul 11, 2019

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