Question

2, 7, 18, 32, ….. form a sequence with a specific pattern. The 7th term of the sequence is what?

Answer 1

To find the 7th term of the sequence 2, 7, 18, 32, we first need to determine the pattern. Let's examine the differences between the terms:

1. The first term is 2.
2. The second term is 7. The difference from the first term is:
\[
7 - 2 = 5
\]
3. The third term is 18. The difference from the second term is:
\[
18 - 7 = 11
\]
4. The fourth term is 32. The difference from the third term is:
\[
32 - 18 = 14
\]

Now let's summarize the differences we have found so far:
- \(a_1 = 2\)
- \(a_2 - a_1 = 5\)
- \(a_3 - a_2 = 11\)
- \(a_4 - a_3 = 14\)

Next, we can find the differences of the differences:
1. \(11 - 5 = 6\)
2. \(14 - 11 = 3\)

From this, we see that the second differences do not appear to be constant, but we can see that the sequence of first differences is: 5, 11, 14. So we can calculate the next difference. Notice the increase by values \(6\) and \(3\). Let’s assume a quadratic form for the sequence. Thus, we will compute a few more terms directly:

We could try finding the next term by continuing the first differences' pattern:
- The differences we have are: \(5, 11, 14\).
- We can deduce that the pattern consists of adding the next difference \(5 + 6 = 11\), then \(11 + 3 = 14\) and keeping the ideas as such.

If we add \(14 + 3 = 17\):

So the next difference (the 5th term) would be \(14 + 17 = 31\):
\[
32 + 31 = 63
\]

Next, we’d now calculate \(63 + 17 = 80\) for the 6th term. Continuing forward in the same logic yields:

Continuing the pattern forward, we see:
\[
80 + 31 = 111 (7th term)
\]

Thus, the 7th term of the sequence is:
\[
\boxed{111}
\]