Question

What will be the next digit in the sentence 1, 1, 2, 3, 5, 8, 13, 21, 34?
Please show me your way of thinking :)

Answer 1

It's Fibonacci :

so you have 
1, 1, 2, 3, 5, 8, 13, 21, 34, 55

Basically, to find out the number:

13=5+8
21=13+8
34=21+13
55=34+21

The next number is the sum of the previous two.

Answer 2

$This\ is\ a\ Fibonacci\ sequence:\\1;\ 1;\ 2;\ 3;\ 5;\ 8;\ 13;\ 21;\ 34;...\\\\a_1=1\\a_2=1\\a_3=a_1+a_2=1+1=2\\a_4=a_2+a_3=1+2=3\\a_5=a_3+a_4=2+3=5\\a_6=a_4+a_5=3+5=8\\a_7=a_5+a_6=5+8=13\\a_8=a_6+a_7=8+13=21\\a_9=a_7+a_8=13+21=34\\a_{10}=a_8+a_9=21+34=55\\a_{11}=a_9+a_{10}=34+55=89\\a_{12}=a_{10}+a_{11}=55+89=144\\\vdots$


$a_n=a_{n-2}+a_{n-1}$-The next number is the sum of the previous two.