Find an explicit rule for the nth term of the sequence 2, 8, 32, 128

To find an explicit rule for the nth term of the sequence 2, 8, 32, 128, we first observe the pattern in the numbers provided.

The numbers can be rewritten as follows:

• 2 = 2¹
• 8 = 2³
• 32 = 2⁵
• 128 = 2⁷

Next, we notice that the exponents form an arithmetic sequence: 1, 3, 5, 7.

The exponents increase by 2 each time. In general terms, we can express the n-th term related to the exponent as:

Exponent for nth term = 2n – 1.

Thus, the nth term of the sequence can be written as:

T(n) = 2^{(2n – 1)}

This formula holds for n = 1, 2, 3, … and produces 2, 8, 32, and 128 for the first four terms. Therefore, the explicit rule for the nth term of the sequence is:

T(n) = 2^{(2n – 1)}.