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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.
Solution
$$a_n + b_n$$ for even n = $$p*b_{n-1} + q*b_{n-1}$$
= $$(p+q)*b_{n-1}$$
$$b_{n-1} = q*a_{n-2} = qp*b_{n-3}$$
= $$q^2*p*a_{n-4} = q^2p^2*b_{n-5}$$
.
.
= $$(qp)^{n/2-1}*b_1 = (qp)^{n/2-1}*q$$
So, $$a_n + b_n$$ = $$q(pq)^{(n/2) -1}(p+q)$$
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