What number should be added to each of the numbers 3/5 13 and 19 so that the resulting number may be in proportion ?

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An Arithmetic Sequence is made by adding the same value each time.

This sequence has a difference of 3 between each number.
The pattern is continued by adding 3 to the last number each time, like this:

This sequence has a difference of 5 between each number.
The pattern is continued by adding 5 to the last number each time, like this:

The value added each time is called the "common difference"

What is the common difference in this example?

Answer: The common difference is 8

The common difference could also be negative:

This common difference is −2
The pattern is continued by subtracting 2 each time, like this:

1739, 1740, 2511, 9771, 9772

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A Geometric Sequence is made by multiplying by the same value each time.

This sequence has a factor of 3 between each number.
The pattern is continued by multiplying by 3 each time, like this:

What we multiply by each time is called the "common ratio".

In the previous example the common ratio was 3:

We can start with any number:

This sequence also has a common ratio of 3, but it starts with 2.

This sequence starts at 1 and has a common ratio of 2.
The pattern is continued by multiplying by 2 each time, like this:

The common ratio can be less than 1:

This sequence starts at 10 and has a common ratio of 0.5 (a half).
The pattern is continued by multiplying by 0.5 each time.

But the common ratio can't be 0, as we get a sequence like 1, 0, 0, 0, 0, 0, 0, ...

658,796, 1741, 10006, 10007

There are also many special sequences, here are some of the most common:

Triangular Numbers

This Triangular Number Sequence is generated from a pattern of dots that form a triangle.

By adding another row of dots and counting all the dots we can find the next number of the sequence:

Square Numbers

They are the squares of whole numbers:

0 (=0×0) 1 (=1×1) 4 (=2×2) 9 (=3×3) 16 (=4×4)

etc...

Cube Numbers

They are the cubes of the counting numbers (they start at 1):

1 (=1×1×1) 8 (=2×2×2) 27 (=3×3×3) 64 (=4×4×4)

etc...

Fibonacci Numbers

The Fibonacci Sequence is found by adding the two numbers before it together. The 2 is found by adding the two numbers before it (1+1) The 21 is found by adding the two numbers before it (8+13) The next number in the sequence above would be 55 (21+34)

Can you figure out the next few numbers?

Other Sequences

There are lots more! You might even think of your own ...

1736, 1737, 3860, 3861, 3862, 1735, 1738