PreCalculus Notes – 9.4 continued RECURSIVE EXPLICIT SEQUENCES: RECURSIVE SEQUENCES: Recursive sequences will give you the first term or so of the sequence. You will have to use the first term to get the second… Ex 1: Write the first 5 terms of the sequence a1 = 2; an+1 = 3 + an. Notice that they already gave us the first term; a1 = 2. When finding terms in the sequence, we ALWAYS start with n = 1 unless otherwise stated. So plug 1 into the equation, then n=2, and so on. Solution: Ex 2: find the first 5 terms of: a1 = 1, a2 = 2, an+2 = anan+1 Here, they have given us the first 2 already. Find the next 3 by plugging in 1, 2, 3 for n. Solution: Ex 3: Fibonacci Sequence: find the first 5 terms of: u1 = 1, u2 = 1, un+2 = un+un+1 Here, they have given us the first 2 already. Find the next 3 by plugging in 1, 2, 3 for n. Solution: Find the first four terms & the eighth term of the recursively defined sequence. 1. a1 = 8, an = an-1-4, for n≥2 2. u1 = -3, un+1 = un +10, for k≥1 3. b1 = 2, bk+1 = 3bk , for k≥1 4. v1 = 0.75, vn = (-2)vn-1 , for n≥2 5. c1 = -2, c2 = 3, ck = ck-2 +ck-1, for k≥3 Find a recursive rule for the nth term. 6. 6, 10, 14, 18, … 7. -4, 1, 6, 11, … 8. -5, -2, 1, 4, … 9. -7, 4, 15, 26, … 10. The fifth and ninth terms of an arithmetic sequence are -5 and -17, respectively. Find a recursive rule for the nth term.

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