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