Question

A computer is infected with the Sasser virus. Assume that it infects 20 other computers within 5 minutes; and that these PCs and servers each infect 20 more machines within another five minutes, etc. How long until 100 million computers are infected?

Answer 1

This is to be done using logarithms

Answer 2

initially only 1 computer is infected
then, after 5 minutes, 20 more are infected (so 1+20 altogether)
then, after another 5 minutes, the initial one infects 20 more and each of the other 20 infect 20 more (so 1+20+20^2)
we therefore have a geometric series representing the number of infected computers after 'n' lots of 5 minutes given by:\[1+20+20^{2}+20^{3}+...+20^{n}\]
the sum of such a series is given by:\[a+ar+ar^{2}+ar^{3}+...+ar^{n}=(r^{n+1}-1)/(r-1)\]
so for your scenario, we have a=1, r=20 and get:\[(20^{n+1}-1)/(20-1)=(20^{n+1}-1)/19\]
you need to know when this will exceed 100,000,000 computers, so we get the equation:\[(20^{n+1}-1)/19\ge10^{8}\]\[20^{n+1}-1\ge19*10^{8}\]\[20^{n+1}\ge1+19*10^{8}\]now take logs of both sides to get:\[\log 20^{n+1}\ge\log (1+19*10^{8})\]\[(n+1)\log 20\ge\log (1+19*10^{8})\]\[n+1\ge\log (1+19*10^{8})/\log 20=9.2787536011814/1.30102999566398=7.1318...\]\[n\ge7.1318...-1\]\[n\ge6.1318...\]
therefore n has to be 7. so it will take 7*5 = 35 minutes to infect at least 100 million computers