1. **State the problem:** We are given the sequence 2, 6, 12, 20, 30 and asked to find the next number. 2. **Identify the pattern:** Let's look at the differences between consecutive terms: $$6 - 2 = 4$$ $$12 - 6 = 6$$ $$20 - 12 = 8$$ $$30 - 20 = 10$$ 3. **Analyze the differences:** The differences are 4, 6, 8, 10 which increase by 2 each time. 4. **Predict the next difference:** Following the pattern, the next difference should be: $$10 + 2 = 12$$ 5. **Calculate the next term:** Add this difference to the last term: $$30 + 12 = 42$$ 6. **Conclusion:** The next number in the sequence is **42**. **Additional insight:** The sequence can also be expressed as $n(n+1)$ for $n=1,2,3,4,5$: - $1 \times 2 = 2$ - $2 \times 3 = 6$ - $3 \times 4 = 12$ - $4 \times 5 = 20$ - $5 \times 6 = 30$ So the next term for $n=6$ is: $$6 \times 7 = 42$$