Suppose now that p > q and a1 < p − q. This implies that an < p − q for all n (you do not needto show that, but it can be done for example by using induction, or in other ways).Find the limit of limn→∞ an. What does this mean for the population?Hint: Show that {an}∞n=1 is increasing and use a known theorem together with your result from part (a).

The population size of a population of leopard geckos under certain circumstances can be modeled by the sequence{an}∞n=1 wherean+1 = panan + qwhere p and q are positive constants. an is the number of geckos in the population after n years. The size of thepopulation in the long term depends on the relative values of p and q, as well as on the initial population a1.(a) (3 marks) Zara writes down the following argument:Zara’s reasoning: If the sequence {an}∞n=1 converges, suppose it converges to some number L. Inthat case:limn→∞ an = L and also limn→∞ an+1 = LSince all the terms are positive, we must have L > 0. Therefore:L = limn→∞ an+1 = limn→∞panan + q = limn→∞p1 + q/an= p1 + q/LTherefore, if the sequence converges, then the limit L satisfies L = p1 + q/L .There is ONE error is Zara’s reasoning. Find the error, then solve for L to find all possible values ofL.(b) (3 marks) Suppose that p < q. Find the limit of limn→∞ an. What does this mean for the leopard geckopopulation?Hint: First show that an+1 < pq an, then use a known theorem together with your result from part (a).(c) (3 marks) Suppose now that p > q and a1 < p − q. This implies that an < p − q for all n (you do not needto show that, but it can be done for example by using induction, or in other ways).Find the limit of limn→∞ an. What does this mean for the population?Hint: Show that {an}∞n=1 is increasing and use a known theorem together with your result from part (a).

The population size of a population of leopard geckos under certain circumstances can be modeled by the sequence{an}∞n=1 wherean+1 = panan + qwhere p and q are positive constants. an is the number of geckos in the population after n years. The size of thepopulation in the long term depends on the relative values of p and q, as well as on the initial population a1.(a) (3 marks) Zara writes down the following argument:Zara’s reasoning: If the sequence {an}∞n=1 converges, suppose it converges to some number L. Inthat case:limn→∞ an = L and also limn→∞ an+1 = LSince all the terms are positive, we must have L > 0. Therefore:L = limn→∞ an+1 = limn→∞panan + q = limn→∞p1 + q/an= p1 + q/LTherefore, if the sequence converges, then the limit L satisfies L = p1 + q/L .There is ONE error is Zara’s reasoning. Find the error, then solve for L to find all possible values ofL.(b) (3 marks) Suppose that p < q. Find the limit of limn→∞ an. What does this mean for the leopard geckopopulation?Hint: First show that an+1 < pq an, then use a known theorem together with your result from part (a).

Show mathematically that lim t → ∞  A(t) exists if p > 1 but does not exist if 0 < p < 1.

lim n → ∞ an+1an = 7

What do you know can limit a population's growth?