Let the position count be ii
Let any term be a_iai
Let total count be nn
So we have a_1+a_2+a_3+...+a_n" "->" "4+8+12+...+312a1+a2+a3+...+an → 4+8+12+...+312
Notice that:
8-4=4
12-8=4
So really this is the sum of the 4 times table
a_1=1xx4=4
a_2=2xx4=8
a_3=3xx4=13
a_n=nxx4=312 => n= 312/4 = 78
color(green)((1color(magenta)(xx4))+(2color(magenta)(xx4))+(3color(magenta)(xx4))+...+(78color(magenta)(xx4)))
color(magenta)(4)color(green)((1+2+3+..+78))
'~~~~~~~~~~~~~~~~~~~
If you really wish to use mathematical notation
Given that the term Sigma means sum of and Sigma_(i=1)^n a_i color(white)(.) means sum of a_1+a_2+...+a_n
Then what we really have is color(magenta)(4xx)color(green)(Sigma_(i=1)^78(i))
The sum of color(green)(1+2+3+...+78)" is count"xx"mean value"
So we have
color(green)(color(magenta)(4xx)Sigma_(i=1)^78(i))color(blue)(" "->" "color(magenta)(4)[78xx(1+78)/2] = color(magenta)(4xx)3081=12324)
'~~~~~~~~~~~~~~~~~~~~~~~~~~
color(brown)("Demonstrating the principle")
Sigma_(i=1)^3(i) =1+2+3 =6" " ->" " 3xx(1+3)/2 =6
Sigma_(i=1)^4(i) = 1+2+3+4 = 10" "->" "4xx(1+4)/2 = 10
Sigma_(i=1)^5(i)=1+2+3+4+5=15" "->" "5xx(1+5)/2=15
