Please help with this assignment

1. The first 10 Fibonacci numbers modulo 5 are 1, 1, 2, 3, 0, 3, 3, 1, 4, 0. You can see this by finding 1, 1, 2, 3, 5, 8, 13, 21, 34, 55 and reducing each num- ber modulo 5 or (better) doing mod 5 computations such as F5 = F4 + F3 = 3+ 2= 0 mod 5, F6 = Fs + F4 = 0+ 3 = 3 mod 5, etc. (a) Determine Fil, F12, ..., F20 modulo 5. (This will be easier if you do computations modulo 5 rather than working out large Fibonacci numbers and then reducing them.) (b) Prove that Fan is a multiple of 5 for every positive integer n. (Hint: Compute F21 and F22 modulo 5 and use (a).) 2. There is a constant c such that cFn = Fn+1 + Fn-2 for n 2 3. (a) Compute several values of Fn+1 + Fn-2 to determine c. (b) Using the value you found in (a), prove that cFn = Fn+1 + Fn-2 for every positive integer n 2 3. The remaining questions are about a recursively defined sequence Gn defined by G1 = 1, G2 = 3, and Gn = 2Gn-1 + Gn-2 for n 2 3. The next few values are G3 = 7 and G4 = 17. 3. Compute the sequence Gn through n = 10. 4. Prove that Gn is odd for every positive integer n. (Remember "strong" induction.) 5. Prove that >Gai-1 = ; (Gan - 1) for every positive integer n. 6. Look at several examples of G? - Gn+1Gn-1 for n 2 2 and write out a careful, general conjecture. (You do not need to prove the conjecture.)