# Find the next number: 1, 9, 19, 30, 41.5,?

What is the next number in the list, and what is the secret formula for this list?

1, 9, 19, 30, 41.5, ?

My guess is that the next number is:

53.25

I noticed that the difference between the numbers in the sequence was

Add 8, 10, 11, 11.5

Meaning that

The difference between those values was 2, 1, 1/2, so it was going down by 1/2 each time. This makes the next difference 1/4, we add 11.75 to 41.5 to get 53.25.

Assuming that this pattern continues indefinitely,

the $$m$$th term is equal to $$12 m+2^{4-m}-19$$.

• The secret formula was that nth term = [(nₓ₋₁ - nₓ₋₂)/2]+6+nₓ₋₁, but your formula works too! Nice job! – FairOPShotgun Jun 19 at 13:34
• @FairOPShotgun I prefer the formula in this answer. It is "closed-form" (does not require computing previous values). – Parcly Taxel Jun 20 at 11:41

The next point could be -

53.5

This is because -

Given 5 points you can fit a unique quartic (degree 4) polynomial. In this case you get $$\frac{1}{48}n^4-\frac{3}{8}n^3+\frac{131}{48}n^2+\frac{17}{8}n-\frac{7}{2}$$

However, given the above we can see that the best answer to the puzzle is in fact -