Arithemetic series addition

Question

Lets say I have M= 1+2+3+4+5+6+7.... (to infinity)

and I have another sequence,N= 6+14+22+30..... (to infinity)

is it possible to say that N = 4M +2 ?

Or is there another way that I can write this?

Answer 1

Let your sequences be $m=1,2,3,4,5\dots, n=6,14,22,30,\dots$. It is true that $n(i)=4m(i)-2$ Then your sums are $M(i)=\sum_{j=1}^ij=\frac 12j(j+1)$ and $N(i)=\sum_{j=1}^i8j-2=8M(i)-2i$. Even after two terms we have $N(2)=20 \neq 8M(2)-2=22$