# (i)Write out a recursive definition for A(n), the number of bacteria present at the beginning of the nth time period. (ii)Determine time when 100m. Colony of bacteria population starts at 50,000. Every 2 hour interval a reading is taken, and the population tripples.

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Population in initial stage (P1)= 50000

Population after 2 hours (P2)= 50000*3 = 150000

Population after 4 hours (P3)= 150000*3 = 450000

Population after 6 hours (P4) = 450000*3 = 135000

First term (T1) = 50000

Second term (T2) = 150000-50000 = 100000

Third term (T3) = 450000-150000 = 300000 = 100000*3

Fourth term (T4) = 1350000-450000 = 900000 = 100000*3^2

From second term onwards we can see a geometric series with initial term 100000 and common ratio of 3.

So when we need to get the total we need to always add T1 term for our calculation since it is not a part of geometric series.

So if the population in n hours is A(n)

`A(n) = 50000+100000(3^n-1)/(3-1) ` where n>0 and n is even.

`A(n) = 50000+100000(3^n-1)/2`

`A(n) = 50000+50000(3^n-1)`

`A(n) = 50000(1+3^n-1)`

`A(n) = 50000*3^n`

So if A(n) = 100million then;

`100*10^6 = 50000*3^n`

`n = 2000`

So it will take 2000 hours to get population of bacteria to 100 million.

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