If-a-1-1-and-n-1-a-n-1-1-1-n-a-n-find-a-n-

Question Number 147357 by Jamshidbek last updated on 20/Jul/21

If a_{1} = 1 and n ⩾ 1 a_{n + 1} = \frac{1}{1 + n \cdot a_{n}}

find a_{n} = ?

Answered by ArielVyny last updated on 20/Jul/21

a_{1} = 1 = \frac{1}{1 + 0}

a_{2} = \frac{1}{1 + a_{1}} = \frac{1}{1 + 1} (n = 1)

a_{3} = \frac{1}{1 + 2 a_{2}} = \frac{1}{1 + 2 \frac{1}{1 + 1}} (n = 2)

a_{4} = \frac{1}{1 + 3 a_{3}} = \frac{1}{1 + 3 (\frac{1}{1 + 2 \frac{1}{1 + 1}})} (n = 3)

a_{5} = \frac{1}{1 + 4 a_{4}} = \frac{1}{1 + 4 (\frac{1}{1 + 3 (\frac{1}{1 + 2 \frac{1}{1 + 1}})})} (n = 4)

a_{n} = \frac{1}{1 + {n a}_{n - 1}} = \frac{1}{1 + n (\frac{1}{1 + (n - 1) (\frac{1}{1 + (n - 2) \frac{1}{1 + 1}}) \dots (\frac{1}{1 + (n - k) \frac{1}{1 + 1}})}}

a_{n} = \frac{1}{1 + n (\frac{1}{1 + \underset{k = 1}{\prod^{n}} (\frac{1}{1 + (n - k) \frac{1}{1 + 1}})})}