Question

2) Write an explicit formula for the sequence
\[2, 9, 16,\unicode{0x2026}\].
\[d(n)=\]

Answer 1

To find an explicit formula for the sequence \(2, 9, 16, \ldots\), we first look for a pattern in the numbers.

1. We note the numbers in the sequence:
\[
a_1 = 2, \quad a_2 = 9, \quad a_3 = 16
\]

2. Next, let's calculate the differences between consecutive terms:
\[
a_2 - a_1 = 9 - 2 = 7
\]
\[
a_3 - a_2 = 16 - 9 = 7
\]

The differences are constant, indicating that this is an arithmetic sequence with a common difference \(d = 7\).

3. The general formula for the nth term of an arithmetic sequence is given by:
\[
a_n = a_1 + (n-1) \cdot d
\]

where \(a_1\) is the first term, \(d\) is the common difference, and \(n\) is the term number.

4. Substituting the values we have:
- \(a_1 = 2\)
- \(d = 7\)

The formula becomes:
\[
a_n = 2 + (n-1) \cdot 7
\]

5. Simplifying this expression:
\[
a_n = 2 + 7n - 7
\]
\[
a_n = 7n - 5
\]

Therefore, the explicit formula for the sequence is:
\[
d(n) = 7n - 5
\]