Maria Callas need to know how many integers ≤ 500 are divisible by (at least one of) 2, 3, or 5
This is inclustion exclusion. First we cound the numbers divisable by 2
![[Graphics:Images/index_gr_1.gif]](Images/index_gr_1.gif)
Similarly for 3 and 5
![[Graphics:Images/index_gr_3.gif]](Images/index_gr_3.gif)
![[Graphics:Images/index_gr_5.gif]](Images/index_gr_5.gif)
![[Graphics:Images/index_gr_7.gif]](Images/index_gr_7.gif)
![[Graphics:Images/index_gr_8.gif]](Images/index_gr_8.gif)
![[Graphics:Images/index_gr_10.gif]](Images/index_gr_10.gif)
![[Graphics:Images/index_gr_12.gif]](Images/index_gr_12.gif)
![[Graphics:Images/index_gr_14.gif]](Images/index_gr_14.gif)
![[Graphics:Images/index_gr_15.gif]](Images/index_gr_15.gif)
![[Graphics:Images/index_gr_17.gif]](Images/index_gr_17.gif)
![[Graphics:Images/index_gr_18.gif]](Images/index_gr_18.gif)
Joan Sutherland would like you tto sum the following. (use the Binomial Theorem)
a:
![[Graphics:Images/index_gr_24.gif]](Images/index_gr_24.gif)
![[Graphics:Images/index_gr_29.gif]](Images/index_gr_29.gif)
Birgit Nilsson would like you to
Solve the difference equation: = 3
+ 4
= 0,
= 5,
= 15.
![[Graphics:Images/index_gr_36.gif]](Images/index_gr_36.gif)
![[Graphics:Images/index_gr_38.gif]](Images/index_gr_38.gif)
![[Graphics:Images/index_gr_40.gif]](Images/index_gr_40.gif)
![[Graphics:Images/index_gr_42.gif]](Images/index_gr_42.gif)
Find a difference equation for the ratios: =
.
Here = 3
+ 4
, as in the previous probem.
from = 3
+ 4
we get
=
= +
= 3 +
= 3 +