Let Xn be the number of successes in n independent Bernoulli trials, each with a success probability p, and

(i) Using the law of total probability and the partition {B1, B2} with B1 = {X1 = 0}, and B2 = {X1 = 1}, prove that the following recursion holds:
an = (1 − p)an−1 + p(1 − an−1), n = 2, 3,…
(ii) Show by induction that the numbers an, n = 1, 2,…, are given by an = 1 2 [1 + (1 − 2p)n], n = 1, 2,…
(iii) Using the result from (ii), deduce the combinatorial identity

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a,, = P(X,, is an even integer).