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
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).
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
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
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!!!
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=65⋮⋮2019a,…,b(a+b2)⋅2019= ?nan,…,bn(an+bn2)⋅n=sn
an= ?
First row:d1=124711…Second Row:d2=1234…Third row:d3=111…
an=(n−10)⋅d1+(n−11)⋅d2+(n−22)⋅d3=(n−10)⋅1+(n−11)⋅1+(n−22)⋅1=1+(n−1)⋅1+(n−2)(n−1)2⋅1⋅1=1+n−1+(n−2)(n−1)2an=n+(n−2)(n−1)2
bn= ?
First row:d1=1361015…Second Row:d2=2345…Third row:d3=111…
bn=(n−10)⋅d1+(n−11)⋅d2+(n−22)⋅d3=(n−10)⋅1+(n−11)⋅2+(n−22)⋅1=1+(n−1)⋅2+(n−2)(n−1)2⋅1⋅1=1+2n−2+(n−2)(n−1)2bn=2n−1+(n−2)(n−1)2
sn=(an+bn2)⋅n=(n+(n−2)(n−1)2+2n−1+(n−2)(n−1)22)⋅n=(3n−1+(n−2)(n−1)2)⋅n=(3n−1+n2−3n+22)⋅nsn=(n2+12)⋅n
The sum of the numbers in row 2019
sn=(n2+12)⋅ns2019=(20192+12)⋅2019s2019=2038181⋅2019s2019=4115087439