A question about $\sum_i^n (A+BX_i)$

Question

$$\sum_{i=1}^n (A+BX_i)= nA +B\sum_{i=1}^n X_i$$

Not sure how to express this question, as english is not my first language.

I know why the A receives an n, but my question is, why does B receive none? B and $X_i$ are separate so shouldn't B receive an N as well?

It is probably aa stupid question, but I couldn't contain my curiosity.

Answer 1

Just try it for $n=3$, for example, and you'll see why: $$ \begin{split} &(A+BX_1)+(A+BX_2)+(A+BX_3) \\&= A+A+A+BX_1+BX_2+BX_3 \\ &= 3A + B(X_1+X_2+X_3) . \end{split} $$

Answer 2

Perhaps an example might help.

Let $A=1$ and $B=4$, $m=2$.

we have

\begin{align} \sum_{i=1}^2(1+4X_i) &= (1+4X_1)+(1+4X_2) \\ &= (1+1) + 4(X_1+X_2) \\ &= 2 + 4(X_1+X_2) \\ &= 2 + B(X_1 + X_2) \end{align}

$2$ is being factorized out of the sum of $X_i$'s.

Answer 3

I will rather explain why the $n$ appears.

The initial expression is

$$\sum_{i=1}^n(A+BX_i)$$

and you can factor out the constant matrices as follows:

$$\sum_{i=1}^n(A+BX_i)=\sum_{i=1}^nA+\sum_{i=1}^nBX_i=A\sum_{i=1}^n1+B\sum_{i=1}^nX_i$$ and obviously,

$$\sum_{i=1}^n1=n.$$