1, 2, 5, 12, 29… Which will be the next number?

 1, 2, 5, 12, 29… Which will be the next number?

👇🏾👇🏾👇🏾👇🏾👇🏾👇🏾👇🏾👇🏾👇 🏾

👇🏾👇🏾👇🏾👇🏾👇🏾👇🏾👇🏾👇🏾👇🏾👇 🏾

To determine the next number in the sequence $1,2,5,12,29,\dots$, we need to find a pattern or rule that generates the sequence.

Let's start by examining the differences between consecutive terms:

$$\mathrm{<b><span\; style="font-size:\; large;">2</span></b>}<b><span\; style="font-size:\; large;">-</span></b>\mathrm{<b><span\; style="font-size:\; large;">1</span></b>}<b><span\; style="font-size:\; large;">=</span></b>\mathrm{<b><span\; style="font-size:\; large;">1</span></b>}$$

$$\mathrm{<b><span\; style="font-size:\; large;">5</span></b>}<b><span\; style="font-size:\; large;">-</span></b>\mathrm{<b><span\; style="font-size:\; large;">2</span></b>}<b><span\; style="font-size:\; large;">=</span></b>\mathrm{<b><span\; style="font-size:\; large;">3</span></b>}$$

$$\mathrm{<b><span\; style="font-size:\; large;">12</span></b>}<b><span\; style="font-size:\; large;">-</span></b>\mathrm{<b><span\; style="font-size:\; large;">5</span></b>}<b><span\; style="font-size:\; large;">=</span></b>\mathrm{<b><span\; style="font-size:\; large;">7</span></b>}$$

$$\mathrm{<b><span\; style="font-size:\; large;">29</span></b>}<b><span\; style="font-size:\; large;">-</span></b>\mathrm{<b><span\; style="font-size:\; large;">12</span></b>}<b><span\; style="font-size:\; large;">=</span></b>\mathrm{<b><span\; style="font-size:\; large;">17</span></b>}$$

Next, we examine the differences of these differences:

$$3-1=2$$

$$7-3=4$$

$$17-7=10$$

We then look at the differences of these second-level differences:

$$4-2=2$$

$$10-4=6$$

It seems the differences of the second-level differences are 2 and 6. Let's examine a hypothesis where the differences at this level follow an arithmetic progression. The differences between consecutive terms in this third level are:

$$\mathrm{<i><b><span\; style="font-size:\; large;">6</span></b></i>}<i><b><span\; style="font-size:\; large;">-</span></b></i>\mathrm{<i><b><span\; style="font-size:\; large;">2</span></b></i>}<i><b><span\; style="font-size:\; large;">=</span></b></i>\mathrm{<i><b><span\; style="font-size:\; large;">4</span></b></i>}$$

Assuming this difference of 4 continues, the next difference in the sequence of third-level differences should be $6+4=10$.

Let's build back up the sequence:

If the next second-level difference is $10+6=16$, then:

If the next first-level difference is $17+16=33$, then:

If the next term is $29+33=62$.

Therefore, the next number in the sequence is:

$$\boxed{{\displaystyle \mathrm{<i><b><span\; style="font-size:\; large;">62</span></b></i>}}}$$