List the first five terms of the sequence given by a_1 = 1, a_(n+1) = (a_n +1) / a_n
_ = Subscript
I hope you were as surprised and pleased as I was, when I noticed that for each a_n, the numerator and denominator are consecutive Fibonacci Numbers F_n/F_(n+1) !
huh? Just plug in the values for n...
a_1 = 1
a_2 = (a_1 + 1)/a_1 = (1+1)/1 = 2
a_3 = (a_2 + 1)/a_2 = (2+1)/2 = 3/2
and so on
Ok, Thank you. So, 1,2,3/2, 4/3, 5/4, 6/5?
no. you just guessed, instead of calculating!
a_4 = (3/2 + 1) / (3/2) = 5/2 / 3/2 = 5/3
To find the first five terms of the sequence given by a_1 = 1 and a_(n+1) = (a_n +1) / a_n, we can use the recursive formula to find each term step by step.
1. The first term, a_1, is given as 1.
2. To find the second term, a_2, we substitute n = 1 into the recursive formula:
a_(1+1) = (a_1 + 1) / a_1
a_2 = (1 + 1) / 1
a_2 = 2 / 1
a_2 = 2
So far, the first two terms of the sequence are 1 and 2.
3. To find the third term, a_3, we substitute n = 2 into the recursive formula:
a_(2+1) = (a_2 + 1) / a_2
a_3 = (2 + 1) / 2
a_3 = 3 / 2
a_3 = 1.5
Now, we have the first three terms: 1, 2, and 1.5.
4. To find the fourth term, a_4, we substitute n = 3 into the recursive formula:
a_(3+1) = (a_3 + 1) / a_3
a_4 = (1.5 + 1) / 1.5
a_4 = 2.5 / 1.5
a_4 ā 1.6667
The first four terms of the sequence are: 1, 2, 1.5, and 1.6667.
5. Finally, we find the fifth term, a_5, by substituting n = 4 into the recursive formula:
a_(4+1) = (a_4 + 1) / a_4
a_5 = (1.6667 + 1) / 1.6667
a_5 ā 1.4
Therefore, the first five terms of the sequence are: 1, 2, 1.5, 1.6667, and 1.4.