Consider a sequence {a} defined recursively by ao = 1 and an = 1+ an-1 when...

 

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Consider a sequence {a} defined recursively by ao = 1 and an = 1+ an-1 when n 1. (a) Compute ao, a1, a2, ..., a10 as fractions in least terms. Do you notice anything about the numerators and denominators? (b) List the decimal expansion of a0, a1, a2, ..., a10. Do you notice any patterns here? (c) We could (but won't) use calculus to show that the sequence a converges to a limit. This limit is the one of the solutions to the equation x = 1+ 1-. The positive solution is 6 = 1+5, and is often called the golden ratio, and the negative solution is = 1-5 (i) Write out the decimal expansions of the first 8 terms of the sequence un = (ii) Compare us to u4+uz. (iii) Use the fact that (iv) Let un = On 1 + 2 = 1 + 1 to show that Vn Z+(un+1 =un un-1 2-1). for n 0, and compute vo, V1, V2, ..., V8. Consider a sequence {a} defined recursively by ao = 1 and an = 1+ an-1 when n 1. (a) Compute ao, a1, a2, ..., a10 as fractions in least terms. Do you notice anything about the numerators and denominators? (b) List the decimal expansion of a0, a1, a2, ..., a10. Do you notice any patterns here? (c) We could (but won't) use calculus to show that the sequence a converges to a limit. This limit is the one of the solutions to the equation x = 1+ 1-. The positive solution is 6 = 1+5, and is often called the golden ratio, and the negative solution is = 1-5 (i) Write out the decimal expansions of the first 8 terms of the sequence un = (ii) Compare us to u4+uz. (iii) Use the fact that (iv) Let un = On 1 + 2 = 1 + 1 to show that Vn Z+(un+1 =un un-1 2-1). for n 0, and compute vo, V1, V2, ..., V8.