This is the third part of the solving number series problems. The first part includes dealing with patterns that contains addition and subtraction and the second part discusses patterns that contains multiplication or division.

In this post, we are going to learn some “alternating sequences.” I put a quote in alternating sequence because in mathematics, it has a slightly different meaning. Note that it is likely that these type of sequence will appear in examinations such as the Civil Service Exam.

Before we continue with the discussion, try to see if you can answer the following questions.

1. 2, -5, 4, -8, 6, -11, 8, -14, ___, ___

2. 4, 7, 12, 15, 20, 23, 28, ____

3. A, 3, D, 8, G, 13, ___, ___

4. , , , , , , ____, _____

**Solutions and Explanations**

*First Sequence: 2, -5, 4, -8, 6, -11, 8, -14, ___, ___*

The first sequence seems hard, but it is actually easy. If you perform addition and subtraction among consecutive terms, you will surely see a pattern (left as an exercise). However, before doing it, notice that the sign of the numbers are alternating: that is, positive, then negative, then positive, and so on.

Now, what if, we separate the two sequences? What if we treat the positive numbers as a sequence, and the negative numbers as another sequence. Well, we just put different colors on them, so it is easy to see the pattern.

2, -5, 4, -8, 6, -11, 8, -14, ___, ___

Do you see now? Can you answer the problem?

As you can see, the red numbers are just increasing by 2 and the blue numbers are decreasing by 3. Therefore, the next numbers are 10 and -17.

*Second Sequence:. 4, 7, 12, 15, 20, 23, 28, ____*

In the second sequence, 4 is increased by 3 to become 7. Then, 7 is increased by 5. The increase in the numbers are also in alternating pattern. So the correct answer is 31 which is equal to 28 + 3.

Further, what is interesting is that the “coloring strategy” that we used in the first sequence can be also used in this sequence. As you can see in the colored numbers below, it becomes two sequences as well. The sequence composed of blue numbers and the other red. In both sequences, the numbers is increased by 8. Since the next number is blue, then it is equal to 23 + 8 = 31.

* 4, 7, 12, 15, 20, 23, 28, ____*

*Third Sequence: A, 3, D, 8, G, 13, ___, ___*

In the third sequence, the answers are already obvious after learning the strategy above. There are two letters in between the letter terms in the sequence (*A, B, C, D, E, F, G, H, I, J*). Further, each number term is 5 greater greater than the previous number term. So, the correct answer answers are *J*, *18*.

* A, 3, D, 8, G, 13, ___, ___*

* *

Fourth Sequence: , , , , , , ____, _____

Sequence 4 is alternating addition. The red numbers as shown in the next figure are added by 1/2 to get the next term while the blue numbers are added by 4. Therefore, the next numbers are 2, 17.

We have done several examples and it is impossible for us to exhaust all patterns, so it is up to you to be able to spot them. The patterns could be different, but the principle of solving them is the same.

In the next post, we are going to look at some of the special sequences that may also appear in the Civil Service Exam.

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