42.8k views
0 votes
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.

User Bisgardo
by
8.5k points

1 Answer

6 votes

Final answer:

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.

User Evil Elf
by
8.9k points
Welcome to QAmmunity.org, where you can ask questions and receive answers from other members of our community.

9.4m questions

12.2m answers

Categories