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Solve the recurrence relation an = an_1 + 8an_2 12 an_3 + 25- (3)&quot;2 + 32 n2 64n, n 2 3, Where do = 130, (11 = 215 and a2 = 260. Write down a...

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Solve the recurrence relation
an = an_1 + 8an_2 — 12 an_3 + 25- (—3)&quot;—2 + 32 n2 — 64n, n 2 3,
Where do = 130, (11 = 215 and a2 = 260. Write down a closed formula for the generating function of the sequence an. (5 marks) TWO sequences an and b“ are related by the identities
bn = (—1)n(n + 1)a0 + (—1)”_1n a1 + - - -+ (—1)2an_1 + an which hold for all n 2 0. Find an expression for the generating function for
the sequence b“ in terms of the generating function for the sequence an. Use your solution to part (a) to prove the identity (2) = f:(—1)k(k + 1) (f: 4: Z) k=0 for all n 2 0, where a is a complex number.

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