When counting forward, we count from the less number to the greater one. Find the difference between two adjacent numbers. Add it to the number that comes before the missing one, or subtract this difference from the number after the unknown one.

When counting backwards, we count from the greater number to the less one. Find out the difference between two adjacent numbers. Subtract it from the number that comes before the missing one, or add the difference to the number after the unknown one. 

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Look at the picture. How are they similar? These are triangles. They all have 3 angles and 3 sides.

How do the shapes on the left differ from the ones on the right? They have different colors and are arranged in another way. The orange triangles are in a row, and green ones are placed randomly.

What if we put green triangles in the line, as is shown in the picture below?

Now all the shapes are in one row. Do these shapes still have some differences? The orange triangles are arranged from the smaller to the bigger one. Green triangles are put randomly and big shapes are mixed with small ones.

Look at the numbers in the orange triangles. They are also in ascending order. Each following number is 1 greater than the previous one. The arrangement of numbers and shapes in a certain order is called a sequence.

There are different types of sequences:

1. In the forward counting sequence, numbers are placed in ascending order. Each following number is greater than the previous one by some identical number. The numbers get greater from left to right.

1 2 3 4 5 6 - the difference between adjacent numbers is 1.

2 4 6 8 10 12 - numbers increase by 2.

10 20 30 40 50 - numbers get greater by 10.

2. In backward counting sequence numbers come from the greater to the less in descending order:

20 19 18 17 16 15 14 13 12 11 10…

3. The sequence of even numbers: 2 4 6 8 10 12 14 16 18 20…

4. The sequence of odd numbers: 1 3 5 7 9 11 13 15 17 19 21…

If you know what type of sequence you deal with, you can continue it and find any missing number. Let’s try to solve the example. We need to fill in the gap in the line of numbers:

90 80 70 ⬜ 50 40 30

1. Is it the forward or backward counting sequence?

The line starts with the number 90 and ends with 30.

30<90, as the numbers decrease from the greater to the less, it is a backward counting sequence.

2. What is the difference between adjacent numbers?

90-80=10 

80-70=10 

50-40=10 

40-30=10

Each following number is 10 less than the preceding one.

3. What number comes before the unknown one? - 70. Subtract 10 from it: 70-10=60.

4. What number is after the missing one? - 50. Add 10 to it: 50+10=60. 5. Complete the sequence using the found number:

90 80 70 60 50 40 30

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