An increasing sequence consists of 4 negative integers and 6 positive

An increasing sequence consists of 4 negative integers and 6 positive integers. Is the sum of the sequence positive?

First be clear about exactly what is being asked about the question. I like to write it down because I've been burnt before by solving the equations correctly but answering something which was not asked.

Is the sum of the sequence positive?

The sequence has 10 numbers four of which are negative and 6 are positive. Represent this information in some logical way. Here is one such way

_ _ _ _ || _ _ _ _ _ _

Numbers on the left of the || are negative and numbers on the right are positive.
All we need to know to solve the question is if the sum on the left is greater than the sum on the right of || . Note that we could compare the average of left to the average of right also. In this question it is not reuired.

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Now we can safely answer the question. Let's start with (2) since it is easier.

(2) The first term of the sequence is -16

Our diagram becomes
-16 _ _ _ || _ _ _ _ _ _

Now if you think about it, we have no other information about the other 9 values. For all we know one of the right values could be 1 trillion making right sum larger. It could also be the case that right values are 1,1,1,1,1,1 which would make positive sum smaller. Hence (2) is insufficient. Cross out B and D
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Now let us consider (1) while momentarily resetting our brain to forget (2)
(1) The difference between any two consecutive negative integers is 5 and the difference between any two consecutive positive integers is 2
our diagram is now
-L -L+5 -L+10 -L+15 || H-10 H-8 H-6 H-4 H-2 H
Where L is the lowest number and His the highest number.
Again we have no information on L and H. If L is minus one trillion and H is 100 then negative sum is higher.
If H is positive one trillion and L is -100 then right sum is obviously higher
Hence (1) is insufficient and we can cross out A

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Now consider (1) and (2) together
(1) The difference between any two consecutive negative integers is 5 and the difference between any two consecutive positive integers is 2

(2) The first term of the sequence is -16
Our diagram becomes

-16 -11 -6 -1 || H-10 H-8 H-6 H-4 H-2 H
We already know for high values of H positive sum must be higher(obviously) So now we need only check for the minimum value of right sum. We can obtain the minimum value of positive sum this way

-16 -11 -6 -1 || 2 4 6 8 10 12

Left sum is -34 and right sum is 42
(Be careful in this step. If right sum was less than or equal to 42 then E would be correct)

Now we know that for whatever value H takes, right sum will always be greater than left sum.
Hence option C is correct.