What explicit formula can be used to generate the infinite sequence 2, 9, 28, 65, 126?

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The sequence you’ve provided is: 2, 9, 28, 65, 126.

To find the explicit formula for this sequence, let’s first examine the terms more closely and look for patterns.

The first few terms can be expressed as follows where n starts from 1:

  • 1st term: 2
  • 2nd term: 9
  • 3rd term: 28
  • 4th term: 65
  • 5th term: 126

Next, to identify a potential pattern, we can compute the values of:

  • 1st term: 1^3 + 1 = 2
  • 2nd term: 2^3 + 1 = 9
  • 3rd term: 3^3 + 1 = 28
  • 4th term: 4^3 + 1 = 65
  • 5th term: 5^3 + 1 = 126

From this examination, we can deduce that each term can be represented by the formula:

n^3 + 1

Thus, the explicit formula to generate the infinite sequence is:

 a_n = n^3 + 1

Where a_n is the nth term of the sequence, and n is a positive integer. This means that if you substitute any positive integer for n, you can generate the corresponding term in the sequence.

For example:

  • If n = 1: a_1 = 1^3 + 1 = 2
  • If n = 2: a_2 = 2^3 + 1 = 9
  • If n = 3: a_3 = 3^3 + 1 = 28
  • If n = 4: a_4 = 4^3 + 1 = 65
  • If n = 5: a_5 = 5^3 + 1 = 126

Therefore, the explicit formula that generates your infinite sequence is a_n = n^3 + 1. Happy calculating!

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