3) Determine a closed form formula for the sequence defined recursively by do = 0, a1 = 1, and for n 2 2, an = 20n-1 + 150,-2. Characteristic Equation: r? = 2r+ 15 12-2--15=0 (r-5) (r+ 3)= 0 Roots : r = - 3, 5 Closed form formula: an = A (- 3 ) " + B. 50 a = A + B = 0 a1 = -30 + 58 = 1 A + B = 0 ( - 3 A + 5 B ) + 3 ( A + B ) = 1 +0 -> 8 8 = 1 -> B= - A = - 3 Hence, an = ( - 3 ) ( - 3) " + ( 3 ) . 50 4) Show the two graphs below are isomorphic by writing down a mapping (in other words, a pairing of the vertices) that is a graph isomorphism. Start with the pairing a - v. Consider the mapping f: 6 - H which maps ! a -> v box ca z dow e zy This is intuitive since G can be considered as a cycle abcdea and Has vxzwyv. It is straightforward to see that the above mapping is bijective , and the corresponding edges are also mapped bijectively . Hence, G and H are isomorphic