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2. The relation Fn+2 = Fn+1 + Fn holds for all integers n and hence so does Fn+3 = Fn+2 + Fn+1. Combine these two formulas to find an expression for Fn+3 in terms of Fn+1 and Fn.
3. Find r, given that Fr = 2F101 + F100.
4. Express F157 + 2F158 in the form Fs.
5. Show the following:
(a) F4 = 3F1 + 2F0.
(b) F5 = 3F2 + 2F1.
6. Add corresponding sides of the formulas of the previous problem and use this to show that F6 = 3F3 + 2F2.
7. Express Fn+4 in terms of Fn+1 and Fn.
8. Find s, given that Fs = 3F200 + 2F199.
9. Find t, given that Ft = 5F317 + 3F316.
10. Find numbers a and b such that Fn+6 = aFn+1 + bFn for all integers n.
11. Show the following:
(a) F0 + F2 + F4 + F6 = F7 - 1.
(b) F0 + F2 + F4 + F6 + F8 = F9 - 1.
(c) F1 + F3 + F5 + F7 = F8.
(d) F1 + F3 + F5 + F7 + F9 = F10.
12. The relation Fn+2 = Fn+1 + Fn can be rewritten as Fn+1 = Fn+2 - Fn. Use this form to find a compact expression for Fa + Fa+2 + Fa+4 + Fa+6 + ... + Fa+2m.
13. Find p, given that Fp = F1 + F3 + F5 + F7 + ... + F701.
14. Find u and v, given that Fu - Fv = F200 + F202 + F204 + ... + F800.
15. Show the following:
(a) F4 = 3F2 - F0.
(b) F5 = 3F3 - F1.
(c) F6 = 3F4 - F2.
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Tuesday, February 17, 1998