The given sequence is:

1, 3, 4, 15, 31, ...

We need to determine whether the sequence is arithmetic, geometric, or neither.

1. Arithmetic Progression Check

In an arithmetic sequence, the difference between consecutive terms is constant.

• Difference between 2nd and 1st terms: $3 - 1 = 2$
• Difference between 3rd and 2nd terms: $4 - 3 = 1$
• Difference between 4th and 3rd terms: $15 - 4 = 11$
• Difference between 5th and 4th terms: $31 - 15 = 16$

The differences are not constant, so the sequence is not arithmetic.

2. Geometric Progression Check

In a geometric sequence, the ratio between consecutive terms is constant.

• Ratio between 2nd and 1st terms: $\frac{3}{1} = 3$
• Ratio between 3rd and 2nd terms: $\frac{4}{3} \approx 1.33$
• Ratio between 4th and 3rd terms: $\frac{15}{4} = 3.75$
• Ratio between 5th and 4th terms: $\frac{31}{15} \approx 2.07$

The ratios are not constant, so the sequence is not geometric.

3. Conclusion

The sequence 1, 3, 4, 15, 31, ... is neither arithmetic nor geometric.

Would you like an explanation on how such sequences could be analyzed further or how to identify other sequence types?

Related Questions:

1. What are the differences between arithmetic and geometric sequences?
2. Can this sequence be part of a recursive pattern?
3. How do you calculate the general term for more complex sequences?
4. What is a Fibonacci-like sequence, and how does it differ?
5. What is the significance of non-arithmetic and non-geometric sequences in math?

Tip: Always look at differences and ratios to identify the type of sequence before making conclusions.