Following, we can consider the number f(n) of perfect matchings in a 3 × 2n grid graph, which is the same as the number of ways to tile a 3 × 2n rectangle with 1 × 2 dominoes. a) Prove that f(0) = 1,f(1) = 3, and that for positive n, f(n) =3f(n−1)+2f(n−2)+ ...+2f(0) b)  Prove(probablyusingtheformulain(a))thatforn>1,f(n)=4f(n−1)−f(n−2). c)Prove by induction (probably using the formula in (b)) that for all naturals n, f(n) = ((1 + 1/√3)(2 + √3)n) + (1 − 1/√3)(2 − √3)n))/2 d)Using any of these formulas, find f(n) for all n ≤ 5

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