2024 DMC Mock 10 Problems/Problem 10
Revision as of 12:08, 22 December 2024 by Pateywatey (talk | contribs) (Created page with "Notice that <math>a_n-a_{n-1}=2(a_{n-1}-a_{n-2})</math>. Since <math>a_2-a_1=1</math>, we get <cmath> a_{100}=(a_{100}-a_{99})+(a_{99}-a_{98})+\dots+(a_2-a_1)=2^{98}+2^{97}+\...")
Notice that 
. Since 
, we get
![\[a_{100}=(a_{100}-a_{99})+(a_{99}-a_{98})+\dots+(a_2-a_1)=2^{98}+2^{97}+\dots+1 = 2^{99}-1\]](http://latex.artofproblemsolving.com/9/9/4/9947d2beb586c5bd299d925367255adee677b077.png)
so 
. Therefore the answer is 
.
