Question

The sequence (sₙ ) is defined by the recursion formula: Sₙ₊₁ = n+1 / 2n Sₙ , S₁ = 1 . Determine an explicit formula for the nth term sₙ in the sequence. Use mathematical induction to prove that your formula is correct.

Answer 1

The explicit formula for the nth term s_n in the sequence is s_n = (n+1)! / (2n * n!). To prove this formula is correct, mathematical induction can be used.

Step-by-step explanation:

To find an explicit formula for the nth term sn in the sequence, we can use the recursion formula:

Sn+1 = (n+1)/(2n) * Sn, S1 = 1

By substituting the values of S1, S2, S3, and so on, we can identify a pattern. The explicit formula for sn can be determined to be sn = (n+1)! / (2n * n!).

To prove that this formula is correct, we can use mathematical induction. We can show that the formula holds for the base case, n = 1, and assume that it holds for n = k. Then we can show that it holds for n = k+1. This completes the proof.