What is 10 - 8 + 6 - 4 + . . .

[VJesus12]

What is $$10-8+6-4+\cdot\ \cdot\ \cdot-(-20)?$$

A. 8
B. 10
C. 12
D. 14
E. 16

The OA is E.

How can I determine the result of the sum? Can I do it without completing the missing numbers?

[ErikaPrepScholar]

We can approach this problem by finding a pattern.

We see that each number decreases by 2, and alternate subtracting and adding. We can start thinking about our equation as subtraction equations that are added together:

(10 - 8) + (6 - 4) ...

So we start with 10 - 8, then we have 6 - 4, then we'll have 2 - 0, and so on. We should notice that each subtraction equations = 2. So we can actually think about our equation like this:

(10 - 8) + (6 - 4) ... = 2 + 2 ...

To get all the way down to -20, we will have 8 different pairs (we can find this by counting down by fours or by dividing 10 - (-20) by 4, rounding down, and adding 1). We add 2 for each pair, so we should get 8*2 = 16.

[Scott@TargetTestPrep]

What is $$10-8+6-4+\cdot\ \cdot\ \cdot-(-20)?$$

A. 8
B. 10
C. 12
D. 14
E. 16

We see that the terms alternate between a plus sign and a minus sign. So, let's group them in pairs:

10 - 8 = 2

6 - 4 = 2
...

The last pair of terms will be:

(-18) - (-20) = 2

We see that every pair of terms sums to 2. Our next question is: How many pairs are there? To answer that, we can just count the first term of each pair, i.e., 10, 6, ..., -18. We see that this is an evenly spaced set with a common difference of 4. Thus, there are (10 - (-18))/4 + 1 = 28/4 + 1 = 8 pairs.

Since there are 8 pairs and each pair add up to 2, the sum of the expression is 8 x 2 = 16.

Answer: E