Question

What is the missing number 1 6 10 30 ?? 53 80

Answer 1

The method of differences gives us `443/15 = 29.5bar(3)`, but that is not a two digit whole number.

Explanation:

Since we are not given any other information about this sequence, the missing value could be anything.

We can analyse it using the method of differences to calculate the missing value on the assumption that it is a polynomial sequence of degree `5` or lower...

Write down the initial sequence, with `x` for the missing value:

`1, 6, 10, 30, x, 53, 80`

Write down the sequence of differences between consecutive terms:

`5, 4, 20, x-30, 53-x, 27`

Write down the sequence of differences of those differences:

`-1, 16, x-5, 83-2x, x-26`

Write down the sequence of differences of those differences:

`17, x-21, 88-3x, 3x-109`

Write down the sequence of differences of those differences:

`x-28, 109-4x, 6x-197`

Write down the sequence of differences of those differences:

`137-5x, 10x-306`

Having taken the difference `5` times, then for a matching quintic sequence, these last two terms must be equal:

`137-5x = 10x-306`

and hence:

`15x=443`

So:

`x = 443/15 = 29.5bar(3)`

This seems unlikely to be the expected solution since it is not a whole `2` digit number, as suggested by the "??".