cat que:An etomologist studying the reproductive patterns of cockroach in an unfavourable env. found that half

Question

the cocroach in any generation failed to reproduce but managed to survive for ten generations while the other half, managed to reproduce doble their no. but failed to survive after that . if teh total population immediately after the 7th generation cockroaches were produced was 8000 . what was the population of the first generation ?

Answer 1

As we don't get beyond 10 generations, the nonproducing cockroaches won't die out here. So we merely have to consider those that reproduce. Let 2n be the number of cockroaches in the first generation; n of these give birth to 2n and then die out, so the next generation will have 3n cockroaches, only n of which can reproduce. And so on. Letting t be the generation number, it follows that the number of cockroaches in that generation is given by (t + 1)n.

So, if at the 7th generation (t = 7) there are 8000, we have that

(7 + 1) n = 8000, ie. n = 1000.

Hence, the initial number of cockroaches is 2000.

Answer 2

If there were say 10 in the first generation, 5 would live to the second while 5 would produce 10 offspring then die. So the second generation would have 15, or 1.5 times as many as the first. So one thing you could do is divide 8000 by 1.5 six times going backwards through the generations. Or, just find 1.5 to the sixth power and divide by that.

Answer 3

if at the beginning of the first generation, there wre 100
then out of that 50 would remain as they wre Remaining 50 would produce 100 and die
so total at the beginning of second generation is 150
so growth rate is 50%compounded annually

let x be the population at the beginning

x(1 + 50/100)^7 = 8000
x(3/2)^7 = 8000
x*2187/128 = 8000
x = 8000*128/2187
468.22