VR - Alternating Terms and Geometric Progression

Geometric Progression in the Differences

Use the following series of numbers to determine what should come next.

13 19 31 55

Following the technique advice of finding the difference between each adjoining number, we would see the following:

6 12 24
13 19 31 55

The series of numbers did not seem to have a great connection whereas, with the differences written in, we can see a clear progression. It’s effectively a two part question – work out the differences and find the following number in that series, before adding it to the last number in the original sequence. The series in italics, 6, 12, 24 will be completed by the number 48 as it is a geometric progression, multiplying the first number by two to make the next. Therefore the answer we need is 55 + 48 = 103.

Again, be wary of the mix of arithmetic and geometric progressions. A child who is only looking for the former could use the technique and spot the differences between the numbers, only to decide on the next number having a difference of 30 from the one before. This is the sort of thing the question setters look to exploit and there is almost guaranteed to be a 'wrong answer' that follows that logic.

Alternating Terms

Use the following series of numbers to determine what should come next.

15 6 21 8 27

There are two ways of approaching these questions. Firstly, follow the basic step suggested in the technique tip:

-9 15 -13 -19
15 6 21 8 27

This looks rather scary as there is clearly nothing simple about the progression. If you use logic you would assume that the next step would be to subtract a number from the 27 given that you subtract, add, subtract add... Given that the first two numbers to be subtracted are 9 and 13, the next should presumably be 17. (9 + 4 = 13, 13 + 4 = 17) The answer is therefore 10.

Now that was all very confusing for a child. Here’s where a good tutor would show you the best approach to solve these types of question.

Technique tip:

When you see a series where the numbers do not all increase or decrease, try a different approach. The likelihood is that the first, third, fifth etc will be connected as one series and the second, fourth, sixth etc will be connected as another. Treat each separately and, instead of working out the difference between neighbouring numbers, work out the differences between alternate ones.

Using this advice, we can break down the bigger series into two smaller series:

15 21 27 and 6 8

The first series has numbers increasing by six each time. The second series involves the addition of two each step. We need to find the sixth number and that means we can ignore the first series. The sixth number is just the fourth number plus two. The answer is therefore 10 – and that is hopefully a lot easier than trawling through the alternative way!

If we remove half the question and are left with a simple arithmetic progression, we have a simpler task which will save your child a lot of time. Ensure they watch out for the numbers that increase and then decrease or vice-versa.

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