Instructions

Directions for the following two questions:

Let $$a_1= p$$ and $$b_1 = q$$, where p and q are positive quantities.

Define $$a_n = pb_{n-1} , b_n = qb_{n-1}$$ , for even n > 1. and $$a_n = pa_{n-1} , b_n = qa_{n-1}$$ , for odd n > 1.

Question 14

If p = 1/3 and q = 2/3 , then what is the smallest odd n such that $$a_n+b_n < 0.01$$?

Solution

$$a_{n} + b_{n}$$ (n is odd) = $$p^{\frac{n+1}{2}}*q^{\frac{n-1}{2}} + p^{\frac{n -1}{2}}*q^{\frac{n+1}{2}}$$ = $$(p + q)pq^{\frac{n-1}{2}}$$

Substituting the values of p and q we get

$$a_{n} + b_{n}$$ = $$(\frac{2}{9})^{\frac{n-1}{2}}$$

Now substitute the values of n and check. 

We can see that the lowest value of n for which 

$$a_{n} + b_{n}$$ < .01 is 9


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