Aside from 2, 2, 5, 13, and 28, what number should come next?

 Aside from 2, 2, 5, 13, and 28, what number should come next?👇🏾👇 🏾

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To determine the next number in the sequence $2,2,5,13,28$, we need to identify the pattern or rule governing the progression. One possible approach is to consider how each number relates to its predecessors.

Let's denote the sequence by ${𝑎}_{𝑛}$, where:

• ${𝑎}_1=2$
• ${𝑎}_2=2$
• ${𝑎}_3=5$
• ${𝑎}_4=13$
• ${𝑎}_5=28$

We suspect there might be a recurrence relation. Observing the differences between successive terms:

• ${𝑎}_3-{𝑎}_2=5-2=3$
• ${𝑎}_4-{𝑎}_3=13-5=8$
• ${𝑎}_5-{𝑎}_4=28-13=15$

Now, let's look at the sequence of differences: $3,8,15$.

We further analyze the differences between these differences:

• $8-3=5$
• $15-8=7$

Thus, the second-order differences are $5,7$. This sequence itself suggests an increasing pattern. If the differences between these differences continue to increase by 2 each time, the next difference should be:

• $7+2=9$

Adding this to the last difference in the first-order sequence:

• $15+9=24$

Therefore, the next difference in the sequence should be $24$. Adding this to the last term in the original sequence:

• $28+24=52$

Thus, the next number in the sequence should be $52$.

Therefore, the number that should come next in the sequence $2,2,5,13,28$ is: $\boxed{{\displaystyle 52}}$