To find the next number in the sequence [tex]\(2, 4, 8, 14, 22, 32\)[/tex], let's carefully analyze the pattern in the differences between consecutive terms in the sequence.
1. Calculate the differences between consecutive terms:
- [tex]\(4 - 2 = 2\)[/tex]
- [tex]\(8 - 4 = 4\)[/tex]
- [tex]\(14 - 8 = 6\)[/tex]
- [tex]\(22 - 14 = 8\)[/tex]
- [tex]\(32 - 22 = 10\)[/tex]
2. The differences between consecutive terms are:
- [tex]\(2, 4, 6, 8, 10\)[/tex]
3. Observing the differences, we note that they form a pattern: each difference increases by 2.
- [tex]\(2 + 2 = 4\)[/tex]
- [tex]\(4 + 2 = 6\)[/tex]
- [tex]\(6 + 2 = 8\)[/tex]
- [tex]\(8 + 2 = 10\)[/tex]
4. To find the next difference in the pattern, we add 2 to the last difference:
- [tex]\(10 + 2 = 12\)[/tex]
5. Now, we add this new difference to the last term in the original sequence to find the next term:
- [tex]\(32 + 12 = 44\)[/tex]
Therefore, the next number in the sequence is [tex]\(\boxed{44}\)[/tex].
