Adding summation with a factor

Question

$$2*\sum_{i=0}^{100}{a_k} - \sum_{i=3}^{103}{a_k}$$

So nothing tricky here, i'm trying to write the above as 1 summation, I realize there is more than 1 answer. So I get you want to have the same starting value and ending value in both summation parts to make into 1 sum. To do that you simply add and subtract those extra terms. My problem is that 2 in front of the summation, since the second summation doesn't have a factor of 2 Can I put them together? I mean how i see it consider $(2*3)-3$ is different from $2*(3-3)$ so how can I put these two together. Thank You

Answer 1

I think the current expression is simple enough, but if you wish to have whatever cancellations occur, we could do the following:

$$2\cdot\sum\limits_{i=0}^{100}a_k-\sum\limits_{i=3}^{103}a_k = 2a_0+2a_1+2a_2+2\cdot\sum\limits_{i=3}^{100}a_k-a_{101}-a_{102}-a_{103}-\sum\limits_{i=3}^{100}a_k$$

$$=2a_0+2a_1+2a_2-a_{101}-a_{102}-a_{103}+\sum\limits_{i=3}^{100}a_k$$

Answer 2

$$2\sum_{i=0}^{100}{a_k} - \sum_{i=3}^{103}{a_k} =$$

$$ 2\sum_{i=0}^{100}{a_k} - \sum_{i=0}^{100}{a_{k+3}} = $$

$$ \sum_{i=0}^{100}({2a_k} - {a_{k+3}}) $$